Algebra Esempi

${(\frac{1}{2} (\cos (\frac{π}{16}) + i \sin (\frac{π}{16})))}^{8}$

Passaggio 1

Applica la proprietà distributiva.

${(\frac{1}{2} \cos (\frac{π}{16}) + \frac{1}{2} (i \sin (\frac{π}{16})))}^{8}$

Passaggio 2

$\frac{1}{2}$ e $\cos (\frac{π}{16})$ .

${(\frac{\cos (\frac{π}{16})}{2} + \frac{1}{2} (i \sin (\frac{π}{16})))}^{8}$

Passaggio 3

Moltiplica $\frac{1}{2} (i \sin (\frac{π}{16}))$ .

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Passaggio 3.1

$i$ e $\frac{1}{2}$ .

${(\frac{\cos (\frac{π}{16})}{2} + \frac{i}{2} \sin (\frac{π}{16}))}^{8}$

Passaggio 3.2

$\frac{i}{2}$ e $\sin (\frac{π}{16})$ .

${(\frac{\cos (\frac{π}{16})}{2} + \frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 4

Usa il teorema binomiale.

${(\frac{\cos (\frac{π}{16})}{2})}^{8} + 8 {(\frac{\cos (\frac{π}{16})}{2})}^{7} \frac{i \sin (\frac{π}{16})}{2} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5

Semplifica i termini.

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Passaggio 5.1

Semplifica ciascun termine.

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Passaggio 5.1.1

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{2^{8}} + 8 {(\frac{\cos (\frac{π}{16})}{2})}^{7} \frac{i \sin (\frac{π}{16})}{2} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.2

Eleva $2$ alla potenza di $8$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + 8 {(\frac{\cos (\frac{π}{16})}{2})}^{7} \frac{i \sin (\frac{π}{16})}{2} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.3

Elimina il fattore comune di $2$ .

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Passaggio 5.1.3.1

Scomponi $2$ da $8 {(\frac{\cos (\frac{π}{16})}{2})}^{7}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + 2 (4 {(\frac{\cos (\frac{π}{16})}{2})}^{7}) \frac{i \sin (\frac{π}{16})}{2} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.3.2

Elimina il fattore comune.

Passaggio 5.1.3.3

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + 4 {(\frac{\cos (\frac{π}{16})}{2})}^{7} (i \sin (\frac{π}{16})) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.4

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + 4 \frac{\cos^{7} (\frac{π}{16})}{2^{7}} (i \sin (\frac{π}{16})) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.5

Eleva $2$ alla potenza di $7$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + 4 \frac{\cos^{7} (\frac{π}{16})}{128} (i \sin (\frac{π}{16})) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.6

Elimina il fattore comune di $4$ .

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Passaggio 5.1.6.1

Scomponi $4$ da $128$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + 4 \frac{\cos^{7} (\frac{π}{16})}{4 (32)} (i \sin (\frac{π}{16})) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.6.2

Elimina il fattore comune.

$\frac{\cos^{8} (\frac{π}{16})}{256} + 4 \frac{\cos^{7} (\frac{π}{16})}{4 \cdot 32} (i \sin (\frac{π}{16})) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.6.3

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{\cos^{7} (\frac{π}{16})}{32} (i \sin (\frac{π}{16})) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.7

Moltiplica $\frac{\cos^{7} (\frac{π}{16})}{32} (i \sin (\frac{π}{16}))$ .

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Passaggio 5.1.7.1

$i$ e $\frac{\cos^{7} (\frac{π}{16})}{32}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16})}{32} \sin (\frac{π}{16}) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.7.2

$\frac{i \cos^{7} (\frac{π}{16})}{32}$ e $\sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{6} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.8

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + 28 \frac{\cos^{6} (\frac{π}{16})}{2^{6}} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.9

Eleva $2$ alla potenza di $6$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + 28 \frac{\cos^{6} (\frac{π}{16})}{64} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.10

Elimina il fattore comune di $4$ .

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Passaggio 5.1.10.1

Scomponi $4$ da $28$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + 4 (7) \frac{\cos^{6} (\frac{π}{16})}{64} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.10.2

Scomponi $4$ da $64$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + 4 \cdot 7 \frac{\cos^{6} (\frac{π}{16})}{4 \cdot 16} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.10.3

Elimina il fattore comune.

Passaggio 5.1.10.4

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + 7 \frac{\cos^{6} (\frac{π}{16})}{16} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.11

$7$ e $\frac{\cos^{6} (\frac{π}{16})}{16}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{7 \cos^{6} (\frac{π}{16})}{16} {(\frac{i \sin (\frac{π}{16})}{2})}^{2} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.12

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.12.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{7 \cos^{6} (\frac{π}{16})}{16} \cdot \frac{{(i \sin (\frac{π}{16}))}^{2}}{2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.12.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{7 \cos^{6} (\frac{π}{16})}{16} \cdot \frac{i^{2} \sin^{2} (\frac{π}{16})}{2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.13

Combina.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{7 \cos^{6} (\frac{π}{16}) (i^{2} \sin^{2} (\frac{π}{16}))}{16 \cdot 2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.14

Semplifica il numeratore.

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Passaggio 5.1.14.1

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{7 \cos^{6} (\frac{π}{16}) \cdot - 1 \sin^{2} (\frac{π}{16})}{16 \cdot 2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.14.2

Raccogli gli esponenti.

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Passaggio 5.1.14.2.1

Metti in evidenza il valore negativo.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- (7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16}))}{16 \cdot 2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.14.2.2

Moltiplica $7$ per $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- 7 (\cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16}))}{16 \cdot 2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- 7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{16 \cdot 2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.15

Semplifica il denominatore.

Tocca per altri passaggi...

Passaggio 5.1.15.1

Riscrivi $16$ come $2^{4}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- 7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{2^{4} \cdot 2^{2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.15.2

Usa la regola della potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- 7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{2^{4 + 2}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.15.3

Somma $4$ e $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- 7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{2^{6}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.16

Eleva $2$ alla potenza di $6$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} + \frac{- 7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.17

Sposta il negativo davanti alla frazione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{5} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.18

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 56 \frac{\cos^{5} (\frac{π}{16})}{2^{5}} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.19

Eleva $2$ alla potenza di $5$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 56 \frac{\cos^{5} (\frac{π}{16})}{32} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.20

Elimina il fattore comune di $8$ .

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Passaggio 5.1.20.1

Scomponi $8$ da $56$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 8 (7) \frac{\cos^{5} (\frac{π}{16})}{32} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.20.2

Scomponi $8$ da $32$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 8 \cdot 7 \frac{\cos^{5} (\frac{π}{16})}{8 \cdot 4} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.20.3

Elimina il fattore comune.

Passaggio 5.1.20.4

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + 7 \frac{\cos^{5} (\frac{π}{16})}{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.21

$7$ e $\frac{\cos^{5} (\frac{π}{16})}{4}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16})}{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{3} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.22

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.22.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16})}{4} \cdot \frac{{(i \sin (\frac{π}{16}))}^{3}}{2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.22.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16})}{4} \cdot \frac{i^{3} \sin^{3} (\frac{π}{16})}{2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.23

Combina.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16}) (i^{3} \sin^{3} (\frac{π}{16}))}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.24

Semplifica il numeratore.

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Passaggio 5.1.24.1

Metti in evidenza $i^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16}) (i^{2} \cdot i) \sin^{3} (\frac{π}{16})}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.24.2

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16}) (- 1 \cdot i) \sin^{3} (\frac{π}{16})}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.24.3

Riscrivi $- 1 i$ come $- i$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{7 \cos^{5} (\frac{π}{16}) \cdot - 1 i \sin^{3} (\frac{π}{16})}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.24.4

Raccogli gli esponenti.

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Passaggio 5.1.24.4.1

Metti in evidenza il valore negativo.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- (7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16}))}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.24.4.2

Moltiplica $7$ per $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- 7 (\cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16}))}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- 7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{4 \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.25

Semplifica il denominatore.

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Passaggio 5.1.25.1

Riscrivi $4$ come $2^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- 7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{2^{2} \cdot 2^{3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.25.2

Usa la regola della potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- 7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{2^{2 + 3}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.25.3

Somma $2$ e $3$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- 7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{2^{5}} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.26

Eleva $2$ alla potenza di $5$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{- 7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.27

Sposta il negativo davanti alla frazione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 70 {(\frac{\cos (\frac{π}{16})}{2})}^{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.28

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 70 \frac{\cos^{4} (\frac{π}{16})}{2^{4}} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.29

Eleva $2$ alla potenza di $4$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 70 \frac{\cos^{4} (\frac{π}{16})}{16} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.30

Elimina il fattore comune di $2$ .

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Passaggio 5.1.30.1

Scomponi $2$ da $70$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 2 (35) \frac{\cos^{4} (\frac{π}{16})}{16} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.30.2

Scomponi $2$ da $16$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 2 \cdot 35 \frac{\cos^{4} (\frac{π}{16})}{2 \cdot 8} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.30.3

Elimina il fattore comune.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 2 \cdot 35 \frac{\cos^{4} (\frac{π}{16})}{2 \cdot 8} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.30.4

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + 35 \frac{\cos^{4} (\frac{π}{16})}{8} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.31

$35$ e $\frac{\cos^{4} (\frac{π}{16})}{8}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16})}{8} {(\frac{i \sin (\frac{π}{16})}{2})}^{4} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.32

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.32.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16})}{8} \cdot \frac{{(i \sin (\frac{π}{16}))}^{4}}{2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.32.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16})}{8} \cdot \frac{i^{4} \sin^{4} (\frac{π}{16})}{2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.33

Combina.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) (i^{4} \sin^{4} (\frac{π}{16}))}{8 \cdot 2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.34

Semplifica il numeratore.

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Passaggio 5.1.34.1

Riscrivi $i^{4}$ come $1$ .

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Passaggio 5.1.34.1.1

Riscrivi $i^{4}$ come ${(i^{2})}^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) {(i^{2})}^{2} \sin^{4} (\frac{π}{16})}{8 \cdot 2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.34.1.2

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) {(- 1)}^{2} \sin^{4} (\frac{π}{16})}{8 \cdot 2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.34.1.3

Eleva $- 1$ alla potenza di $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \cdot 1 \sin^{4} (\frac{π}{16})}{8 \cdot 2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.34.2

Moltiplica $35$ per $1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{8 \cdot 2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.35

Semplifica il denominatore.

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Passaggio 5.1.35.1

Riscrivi $8$ come $2^{3}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{2^{3} \cdot 2^{4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.35.2

Usa la regola della potenza $a^{m} a^{n} = a^{m + n}$ per combinare gli esponenti.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{2^{3 + 4}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.35.3

Somma $3$ e $4$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{2^{7}} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.36

Eleva $2$ alla potenza di $7$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 56 {(\frac{\cos (\frac{π}{16})}{2})}^{3} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.37

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 56 \frac{\cos^{3} (\frac{π}{16})}{2^{3}} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.38

Eleva $2$ alla potenza di $3$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 56 \frac{\cos^{3} (\frac{π}{16})}{8} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.39

Elimina il fattore comune di $8$ .

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Passaggio 5.1.39.1

Scomponi $8$ da $56$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 8 (7) \frac{\cos^{3} (\frac{π}{16})}{8} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.39.2

Elimina il fattore comune.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 8 \cdot 7 \frac{\cos^{3} (\frac{π}{16})}{8} {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.39.3

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) {(\frac{i \sin (\frac{π}{16})}{2})}^{5} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.40

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.40.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{{(i \sin (\frac{π}{16}))}^{5}}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.40.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{i^{5} \sin^{5} (\frac{π}{16})}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.41

Semplifica il numeratore.

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Passaggio 5.1.41.1

Metti in evidenza $i^{4}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{i^{4} i \sin^{5} (\frac{π}{16})}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.41.2

Riscrivi $i^{4}$ come $1$ .

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Passaggio 5.1.41.2.1

Riscrivi $i^{4}$ come ${(i^{2})}^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{{(i^{2})}^{2} i \sin^{5} (\frac{π}{16})}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.41.2.2

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{{(- 1)}^{2} i \sin^{5} (\frac{π}{16})}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.41.2.3

Eleva $- 1$ alla potenza di $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{1 i \sin^{5} (\frac{π}{16})}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.41.3

Moltiplica $i$ per $1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{i \sin^{5} (\frac{π}{16})}{2^{5}} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.42

Eleva $2$ alla potenza di $5$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + 7 \cos^{3} (\frac{π}{16}) \frac{i \sin^{5} (\frac{π}{16})}{32} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.43

Moltiplica $7 \cos^{3} (\frac{π}{16}) \frac{i \sin^{5} (\frac{π}{16})}{32}$ .

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Passaggio 5.1.43.1

$\frac{i \sin^{5} (\frac{π}{16})}{32}$ e $7$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{i \sin^{5} (\frac{π}{16}) \cdot 7}{32} \cos^{3} (\frac{π}{16}) + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.43.2

$\frac{i \sin^{5} (\frac{π}{16}) \cdot 7}{32}$ e $\cos^{3} (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{i \sin^{5} (\frac{π}{16}) \cdot 7 \cos^{3} (\frac{π}{16})}{32} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.44

Sposta $7$ alla sinistra di $i \sin^{5} (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 \cdot (i \sin^{5} (\frac{π}{16})) \cos^{3} (\frac{π}{16})}{32} + 28 {(\frac{\cos (\frac{π}{16})}{2})}^{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.45

Applica la regola del prodotto a $\frac{\cos (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 28 \frac{\cos^{2} (\frac{π}{16})}{2^{2}} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.46

Eleva $2$ alla potenza di $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 28 \frac{\cos^{2} (\frac{π}{16})}{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.47

Elimina il fattore comune di $4$ .

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Passaggio 5.1.47.1

Scomponi $4$ da $28$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 4 (7) \frac{\cos^{2} (\frac{π}{16})}{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.47.2

Elimina il fattore comune.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 4 \cdot 7 \frac{\cos^{2} (\frac{π}{16})}{4} {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.47.3

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) {(\frac{i \sin (\frac{π}{16})}{2})}^{6} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.48

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.48.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{{(i \sin (\frac{π}{16}))}^{6}}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.48.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{i^{6} \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.49

Semplifica il numeratore.

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Passaggio 5.1.49.1

Metti in evidenza $i^{4}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{i^{4} i^{2} \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.49.2

Riscrivi $i^{4}$ come $1$ .

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Passaggio 5.1.49.2.1

Riscrivi $i^{4}$ come ${(i^{2})}^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{{(i^{2})}^{2} i^{2} \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.49.2.2

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{{(- 1)}^{2} i^{2} \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.49.2.3

Eleva $- 1$ alla potenza di $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{1 i^{2} \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.49.3

Moltiplica $i^{2}$ per $1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{i^{2} \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.49.4

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{- \sin^{6} (\frac{π}{16})}{2^{6}} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.50

Eleva $2$ alla potenza di $6$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) \frac{- \sin^{6} (\frac{π}{16})}{64} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.51

Sposta il negativo davanti alla frazione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + 7 \cos^{2} (\frac{π}{16}) (- \frac{\sin^{6} (\frac{π}{16})}{64}) + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.52

Moltiplica $7 \cos^{2} (\frac{π}{16}) (- \frac{\sin^{6} (\frac{π}{16})}{64})$ .

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Passaggio 5.1.52.1

Moltiplica $- 1$ per $7$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - 7 \cos^{2} (\frac{π}{16}) \frac{\sin^{6} (\frac{π}{16})}{64} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.52.2

$\frac{\sin^{6} (\frac{π}{16})}{64}$ e $- 7$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + \frac{\sin^{6} (\frac{π}{16}) \cdot - 7}{64} \cos^{2} (\frac{π}{16}) + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.52.3

$\frac{\sin^{6} (\frac{π}{16}) \cdot - 7}{64}$ e $\cos^{2} (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + \frac{\sin^{6} (\frac{π}{16}) \cdot - 7 \cos^{2} (\frac{π}{16})}{64} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.53

Sposta $- 7$ alla sinistra di $\sin^{6} (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} + \frac{- 7 \cdot \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.54

Sposta il negativo davanti alla frazione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \cdot \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 8 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.55

Elimina il fattore comune di $2$ .

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Passaggio 5.1.55.1

Scomponi $2$ da $8$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 2 (4) \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.55.2

Elimina il fattore comune.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 2 \cdot 4 \frac{\cos (\frac{π}{16})}{2} {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.55.3

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) {(\frac{i \sin (\frac{π}{16})}{2})}^{7} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.56

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.56.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{{(i \sin (\frac{π}{16}))}^{7}}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.56.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{i^{7} \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57

Semplifica il numeratore.

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Passaggio 5.1.57.1

Riscrivi $i^{7}$ come $i^{4} (i^{2} \cdot i)$ .

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Passaggio 5.1.57.1.1

Metti in evidenza $i^{4}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{i^{4} i^{3} \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.1.2

Metti in evidenza $i^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{i^{4} (i^{2} \cdot i) \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.2

Riscrivi $i^{4}$ come $1$ .

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Passaggio 5.1.57.2.1

Riscrivi $i^{4}$ come ${(i^{2})}^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{{(i^{2})}^{2} (i^{2} \cdot i) \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.2.2

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{{(- 1)}^{2} (i^{2} \cdot i) \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.2.3

Eleva $- 1$ alla potenza di $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{1 (i^{2} \cdot i) \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.3

Moltiplica $i^{2} \cdot i$ per $1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{i^{2} \cdot i \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.4

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{- 1 \cdot i \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.57.5

Riscrivi $- 1 i$ come $- i$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{- i \sin^{7} (\frac{π}{16})}{2^{7}} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.58

Eleva $2$ alla potenza di $7$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{- i \sin^{7} (\frac{π}{16})}{128} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.59

Elimina il fattore comune di $4$ .

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Passaggio 5.1.59.1

Scomponi $4$ da $4 \cos (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 (\cos (\frac{π}{16})) \frac{- i \sin^{7} (\frac{π}{16})}{128} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.59.2

Scomponi $4$ da $128$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 (\cos (\frac{π}{16})) \frac{- i \sin^{7} (\frac{π}{16})}{4 (32)} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.59.3

Elimina il fattore comune.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + 4 \cos (\frac{π}{16}) \frac{- i \sin^{7} (\frac{π}{16})}{4 \cdot 32} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.59.4

Riscrivi l'espressione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + \cos (\frac{π}{16}) \frac{- i \sin^{7} (\frac{π}{16})}{32} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.60

$\cos (\frac{π}{16})$ e $\frac{- i \sin^{7} (\frac{π}{16})}{32}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + \frac{\cos (\frac{π}{16}) (- i \sin^{7} (\frac{π}{16}))}{32} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.61

Metti in evidenza il valore negativo.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + \frac{- \cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.62

Sposta il negativo davanti alla frazione.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + {(\frac{i \sin (\frac{π}{16})}{2})}^{8}$

Passaggio 5.1.63

Usa la regola della potenza ${(a b)}^{n} = a^{n} b^{n}$ per distribuire l'esponente.

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Passaggio 5.1.63.1

Applica la regola del prodotto a $\frac{i \sin (\frac{π}{16})}{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{{(i \sin (\frac{π}{16}))}^{8}}{2^{8}}$

Passaggio 5.1.63.2

Applica la regola del prodotto a $i \sin (\frac{π}{16})$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{i^{8} \sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.64

Semplifica il numeratore.

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Passaggio 5.1.64.1

Riscrivi $i^{8}$ come ${(i^{4})}^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{{(i^{4})}^{2} \sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.64.2

Riscrivi $i^{4}$ come $1$ .

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Passaggio 5.1.64.2.1

Riscrivi $i^{4}$ come ${(i^{2})}^{2}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{{({(i^{2})}^{2})}^{2} \sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.64.2.2

Riscrivi $i^{2}$ come $- 1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{{({(- 1)}^{2})}^{2} \sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.64.2.3

Eleva $- 1$ alla potenza di $2$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{1^{2} \sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.64.3

Uno elevato a qualsiasi potenza è uno.

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{1 \sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.64.4

Moltiplica $\sin^{8} (\frac{π}{16})$ per $1$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{2^{8}}$

Passaggio 5.1.65

Eleva $2$ alla potenza di $8$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$

Passaggio 5.2

Semplifica con la commutazione.

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Passaggio 5.2.1

Riordina i fattori in $\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$ .

$\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{i \cos (\frac{π}{16}) \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$

Passaggio 5.2.2

Riordina $\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32}$ e $- \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64}$ .

$- \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{i \cos (\frac{π}{16}) \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$

Passaggio 5.2.3

Riordina $- \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32}$ e $\frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128}$ .

$\frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{i \cos (\frac{π}{16}) \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$

Passaggio 5.2.4

Riordina $\frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32}$ e $- \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64}$ .

$- \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{i \cos (\frac{π}{16}) \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$

Passaggio 5.2.5

Riordina $- \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{i \cos (\frac{π}{16}) \sin^{7} (\frac{π}{16})}{32}$ e $\frac{\sin^{8} (\frac{π}{16})}{256}$ .

$\frac{\sin^{8} (\frac{π}{16})}{256} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} + \frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 i \cos^{5} (\frac{π}{16}) \sin^{3} (\frac{π}{16})}{32} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{i \cos (\frac{π}{16}) \sin^{7} (\frac{π}{16})}{32}$