Álgebra Ejemplos

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

Paso 1

Aplica la propiedad distributiva.

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

Paso 2

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

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

Paso 3

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

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Paso 3.1

Combina $i$ y $\frac{1}{2}$ .

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

Paso 3.2

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

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

Paso 4

Usa el teorema del binomio.

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

Paso 5

Simplifica los términos.

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Paso 5.1

Simplifica cada término.

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Paso 5.1.1

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.2

Eleva $2$ a la potencia de $8$ .

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

Paso 5.1.3

Cancela el factor común de $2$ .

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Paso 5.1.3.1

Factoriza $2$ de $8 {(\frac{\cos (\frac{π}{16})}{2})}^{7}$ .

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

Paso 5.1.3.2

Cancela el factor común.

Paso 5.1.3.3

Reescribe la expresión.

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

Paso 5.1.4

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.5

Eleva $2$ a la potencia de $7$ .

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

Paso 5.1.6

Cancela el factor común de $4$ .

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Paso 5.1.6.1

Factoriza $4$ de $128$ .

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

Paso 5.1.6.2

Cancela el factor común.

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

Paso 5.1.6.3

Reescribe la expresión.

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

Paso 5.1.7

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

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Paso 5.1.7.1

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

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

Paso 5.1.7.2

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

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

Paso 5.1.8

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.9

Eleva $2$ a la potencia de $6$ .

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

Paso 5.1.10

Cancela el factor común de $4$ .

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Paso 5.1.10.1

Factoriza $4$ de $28$ .

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

Paso 5.1.10.2

Factoriza $4$ de $64$ .

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

Paso 5.1.10.3

Cancela el factor común.

Paso 5.1.10.4

Reescribe la expresión.

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

Paso 5.1.11

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

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

Paso 5.1.12

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.12.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.12.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.13

Combinar.

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

Paso 5.1.14

Simplifica el numerador.

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Paso 5.1.14.1

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.14.2

Combina exponentes.

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Paso 5.1.14.2.1

Factoriza el negativo.

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

Paso 5.1.14.2.2

Multiplica $7$ por $- 1$ .

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

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

Paso 5.1.15

Simplifica el denominador.

Toca para ver más pasos...

Paso 5.1.15.1

Reescribe $16$ como $2^{4}$ .

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

Paso 5.1.15.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 5.1.15.3

Suma $4$ y $2$ .

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

Paso 5.1.16

Eleva $2$ a la potencia de $6$ .

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

Paso 5.1.17

Mueve el negativo al frente de la fracción.

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

Paso 5.1.18

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.19

Eleva $2$ a la potencia de $5$ .

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

Paso 5.1.20

Cancela el factor común de $8$ .

Toca para ver más pasos...

Paso 5.1.20.1

Factoriza $8$ de $56$ .

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

Paso 5.1.20.2

Factoriza $8$ de $32$ .

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

Paso 5.1.20.3

Cancela el factor común.

Paso 5.1.20.4

Reescribe la expresión.

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

Paso 5.1.21

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

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

Paso 5.1.22

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.22.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.22.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.23

Combinar.

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

Paso 5.1.24

Simplifica el numerador.

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Paso 5.1.24.1

Factoriza $i^{2}$ .

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

Paso 5.1.24.2

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.24.3

Reescribe $- 1 i$ como $- i$ .

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

Paso 5.1.24.4

Combina exponentes.

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Paso 5.1.24.4.1

Factoriza el negativo.

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

Paso 5.1.24.4.2

Multiplica $7$ por $- 1$ .

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

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

Paso 5.1.25

Simplifica el denominador.

Toca para ver más pasos...

Paso 5.1.25.1

Reescribe $4$ como $2^{2}$ .

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

Paso 5.1.25.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 5.1.25.3

Suma $2$ y $3$ .

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

Paso 5.1.26

Eleva $2$ a la potencia de $5$ .

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

Paso 5.1.27

Mueve el negativo al frente de la fracción.

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

Paso 5.1.28

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.29

Eleva $2$ a la potencia de $4$ .

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

Paso 5.1.30

Cancela el factor común de $2$ .

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Paso 5.1.30.1

Factoriza $2$ de $70$ .

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

Paso 5.1.30.2

Factoriza $2$ de $16$ .

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

Paso 5.1.30.3

Cancela el factor común.

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

Paso 5.1.30.4

Reescribe la expresión.

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

Paso 5.1.31

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

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

Paso 5.1.32

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.32.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.32.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.33

Combinar.

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

Paso 5.1.34

Simplifica el numerador.

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Paso 5.1.34.1

Reescribe $i^{4}$ como $1$ .

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Paso 5.1.34.1.1

Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

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

Paso 5.1.34.1.2

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.34.1.3

Eleva $- 1$ a la potencia de $2$ .

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

Paso 5.1.34.2

Multiplica $35$ por $1$ .

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

Paso 5.1.35

Simplifica el denominador.

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Paso 5.1.35.1

Reescribe $8$ como $2^{3}$ .

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

Paso 5.1.35.2

Usa la regla de la potencia $a^{m} a^{n} = a^{m + n}$ para combinar exponentes.

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

Paso 5.1.35.3

Suma $3$ y $4$ .

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

Paso 5.1.36

Eleva $2$ a la potencia de $7$ .

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

Paso 5.1.37

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.38

Eleva $2$ a la potencia de $3$ .

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

Paso 5.1.39

Cancela el factor común de $8$ .

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Paso 5.1.39.1

Factoriza $8$ de $56$ .

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

Paso 5.1.39.2

Cancela el factor común.

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

Paso 5.1.39.3

Reescribe la expresión.

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

Paso 5.1.40

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.40.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.40.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.41

Simplifica el numerador.

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Paso 5.1.41.1

Factoriza $i^{4}$ .

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

Paso 5.1.41.2

Reescribe $i^{4}$ como $1$ .

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Paso 5.1.41.2.1

Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

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

Paso 5.1.41.2.2

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.41.2.3

Eleva $- 1$ a la potencia de $2$ .

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

Paso 5.1.41.3

Multiplica $i$ por $1$ .

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

Paso 5.1.42

Eleva $2$ a la potencia de $5$ .

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

Paso 5.1.43

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

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Paso 5.1.43.1

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

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

Paso 5.1.43.2

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

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

Paso 5.1.44

Mueve $7$ a la izquierda de $i \sin^{5} (\frac{π}{16})$ .

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

Paso 5.1.45

Aplica la regla del producto a $\frac{\cos (\frac{π}{16})}{2}$ .

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

Paso 5.1.46

Eleva $2$ a la potencia de $2$ .

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

Paso 5.1.47

Cancela el factor común de $4$ .

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Paso 5.1.47.1

Factoriza $4$ de $28$ .

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

Paso 5.1.47.2

Cancela el factor común.

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

Paso 5.1.47.3

Reescribe la expresión.

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

Paso 5.1.48

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.48.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.48.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.49

Simplifica el numerador.

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Paso 5.1.49.1

Factoriza $i^{4}$ .

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

Paso 5.1.49.2

Reescribe $i^{4}$ como $1$ .

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Paso 5.1.49.2.1

Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

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

Paso 5.1.49.2.2

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.49.2.3

Eleva $- 1$ a la potencia de $2$ .

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

Paso 5.1.49.3

Multiplica $i^{2}$ por $1$ .

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

Paso 5.1.49.4

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.50

Eleva $2$ a la potencia de $6$ .

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

Paso 5.1.51

Mueve el negativo al frente de la fracción.

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

Paso 5.1.52

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

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Paso 5.1.52.1

Multiplica $- 1$ por $7$ .

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

Paso 5.1.52.2

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

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

Paso 5.1.52.3

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

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

Paso 5.1.53

Mueve $- 7$ a la izquierda de $\sin^{6} (\frac{π}{16})$ .

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

Paso 5.1.54

Mueve el negativo al frente de la fracción.

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

Paso 5.1.55

Cancela el factor común de $2$ .

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Paso 5.1.55.1

Factoriza $2$ de $8$ .

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

Paso 5.1.55.2

Cancela el factor común.

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

Paso 5.1.55.3

Reescribe la expresión.

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

Paso 5.1.56

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.56.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.56.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.57

Simplifica el numerador.

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Paso 5.1.57.1

Reescribe $i^{7}$ como $i^{4} (i^{2} \cdot i)$ .

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Paso 5.1.57.1.1

Factoriza $i^{4}$ .

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

Paso 5.1.57.1.2

Factoriza $i^{2}$ .

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

Paso 5.1.57.2

Reescribe $i^{4}$ como $1$ .

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Paso 5.1.57.2.1

Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

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

Paso 5.1.57.2.2

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.57.2.3

Eleva $- 1$ a la potencia de $2$ .

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

Paso 5.1.57.3

Multiplica $i^{2} \cdot i$ por $1$ .

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

Paso 5.1.57.4

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.57.5

Reescribe $- 1 i$ como $- i$ .

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

Paso 5.1.58

Eleva $2$ a la potencia de $7$ .

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

Paso 5.1.59

Cancela el factor común de $4$ .

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Paso 5.1.59.1

Factoriza $4$ de $4 \cos (\frac{π}{16})$ .

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

Paso 5.1.59.2

Factoriza $4$ de $128$ .

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

Paso 5.1.59.3

Cancela el factor común.

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

Paso 5.1.59.4

Reescribe la expresión.

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

Paso 5.1.60

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

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

Paso 5.1.61

Factoriza el negativo.

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

Paso 5.1.62

Mueve el negativo al frente de la fracción.

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

Paso 5.1.63

Usa la regla de la potencia ${(a b)}^{n} = a^{n} b^{n}$ para distribuir el exponente.

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Paso 5.1.63.1

Aplica la regla del producto a $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Paso 5.1.63.2

Aplica la regla del producto a $i \sin (\frac{π}{16})$ .

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

Paso 5.1.64

Simplifica el numerador.

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Paso 5.1.64.1

Reescribe $i^{8}$ como ${(i^{4})}^{2}$ .

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

Paso 5.1.64.2

Reescribe $i^{4}$ como $1$ .

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Paso 5.1.64.2.1

Reescribe $i^{4}$ como ${(i^{2})}^{2}$ .

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

Paso 5.1.64.2.2

Reescribe $i^{2}$ como $- 1$ .

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

Paso 5.1.64.2.3

Eleva $- 1$ a la potencia de $2$ .

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

Paso 5.1.64.3

Uno elevado a cualquier potencia es uno.

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

Paso 5.1.64.4

Multiplica $\sin^{8} (\frac{π}{16})$ por $1$ .

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

Paso 5.1.65

Eleva $2$ a la potencia de $8$ .

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

Paso 5.2

Simplifica con la conmutatividad.

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Paso 5.2.1

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

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

Paso 5.2.2

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

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

Paso 5.2.3

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

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

Paso 5.2.4

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

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

Paso 5.2.5

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

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