Algebra Beispiele

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

Schritt 1

Wende das Distributivgesetz an.

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

Schritt 2

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

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

Schritt 3

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

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Schritt 3.1

Kombiniere $i$ und $\frac{1}{2}$ .

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

Schritt 3.2

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

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

Schritt 4

Wende den binomischen Lehrsatz an.

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

Schritt 5

Vereinfache Terme.

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Schritt 5.1

Vereinfache jeden Term.

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Schritt 5.1.1

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.2

Potenziere $2$ mit $8$ .

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

Schritt 5.1.3

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 5.1.3.1

Faktorisiere $2$ aus $8 {(\frac{\cos (\frac{π}{16})}{2})}^{7}$ heraus.

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

Schritt 5.1.3.2

Kürze den gemeinsamen Faktor.

Schritt 5.1.3.3

Forme den Ausdruck um.

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

Schritt 5.1.4

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.5

Potenziere $2$ mit $7$ .

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

Schritt 5.1.6

Kürze den gemeinsamen Faktor von $4$ .

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Schritt 5.1.6.1

Faktorisiere $4$ aus $128$ heraus.

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

Schritt 5.1.6.2

Kürze den gemeinsamen Faktor.

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

Schritt 5.1.6.3

Forme den Ausdruck um.

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

Schritt 5.1.7

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

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Schritt 5.1.7.1

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

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

Schritt 5.1.7.2

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

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

Schritt 5.1.8

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.9

Potenziere $2$ mit $6$ .

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

Schritt 5.1.10

Kürze den gemeinsamen Faktor von $4$ .

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Schritt 5.1.10.1

Faktorisiere $4$ aus $28$ heraus.

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

Schritt 5.1.10.2

Faktorisiere $4$ aus $64$ heraus.

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

Schritt 5.1.10.3

Kürze den gemeinsamen Faktor.

Schritt 5.1.10.4

Forme den Ausdruck um.

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

Schritt 5.1.11

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

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

Schritt 5.1.12

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

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Schritt 5.1.12.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.12.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.13

Kombinieren.

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

Schritt 5.1.14

Vereinfache den Zähler.

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Schritt 5.1.14.1

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.14.2

Kombiniere Exponenten.

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Schritt 5.1.14.2.1

Faktorisiere das negative Vorzeichen heraus.

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

Schritt 5.1.14.2.2

Mutltipliziere $7$ mit $- 1$ .

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

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

Schritt 5.1.15

Vereinfache den Nenner.

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Schritt 5.1.15.1

Schreibe $16$ als $2^{4}$ um.

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

Schritt 5.1.15.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

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

Schritt 5.1.15.3

Addiere $4$ und $2$ .

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

Schritt 5.1.16

Potenziere $2$ mit $6$ .

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

Schritt 5.1.17

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 5.1.18

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.19

Potenziere $2$ mit $5$ .

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

Schritt 5.1.20

Kürze den gemeinsamen Faktor von $8$ .

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Schritt 5.1.20.1

Faktorisiere $8$ aus $56$ heraus.

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

Schritt 5.1.20.2

Faktorisiere $8$ aus $32$ heraus.

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

Schritt 5.1.20.3

Kürze den gemeinsamen Faktor.

Schritt 5.1.20.4

Forme den Ausdruck um.

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

Schritt 5.1.21

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

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

Schritt 5.1.22

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

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Schritt 5.1.22.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.22.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.23

Kombinieren.

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

Schritt 5.1.24

Vereinfache den Zähler.

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Schritt 5.1.24.1

Faktorisiere $i^{2}$ aus.

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

Schritt 5.1.24.2

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.24.3

Schreibe $- 1 i$ als $- i$ um.

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

Schritt 5.1.24.4

Kombiniere Exponenten.

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Schritt 5.1.24.4.1

Faktorisiere das negative Vorzeichen heraus.

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

Schritt 5.1.24.4.2

Mutltipliziere $7$ mit $- 1$ .

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

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

Schritt 5.1.25

Vereinfache den Nenner.

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Schritt 5.1.25.1

Schreibe $4$ als $2^{2}$ um.

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

Schritt 5.1.25.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

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

Schritt 5.1.25.3

Addiere $2$ und $3$ .

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

Schritt 5.1.26

Potenziere $2$ mit $5$ .

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

Schritt 5.1.27

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 5.1.28

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.29

Potenziere $2$ mit $4$ .

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

Schritt 5.1.30

Kürze den gemeinsamen Faktor von $2$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1.30.1

Faktorisiere $2$ aus $70$ heraus.

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

Schritt 5.1.30.2

Faktorisiere $2$ aus $16$ heraus.

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

Schritt 5.1.30.3

Kürze den gemeinsamen Faktor.

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

Schritt 5.1.30.4

Forme den Ausdruck um.

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

Schritt 5.1.31

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

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

Schritt 5.1.32

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1.32.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.32.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.33

Kombinieren.

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

Schritt 5.1.34

Vereinfache den Zähler.

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1.34.1

Schreibe $i^{4}$ als $1$ um.

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1.34.1.1

Schreibe $i^{4}$ als ${(i^{2})}^{2}$ um.

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

Schritt 5.1.34.1.2

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.34.1.3

Potenziere $- 1$ mit $2$ .

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

Schritt 5.1.34.2

Mutltipliziere $35$ mit $1$ .

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

Schritt 5.1.35

Vereinfache den Nenner.

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1.35.1

Schreibe $8$ als $2^{3}$ um.

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

Schritt 5.1.35.2

Wende die Exponentenregel $a^{m} a^{n} = a^{m + n}$ an, um die Exponenten zu kombinieren.

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

Schritt 5.1.35.3

Addiere $3$ und $4$ .

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

Schritt 5.1.36

Potenziere $2$ mit $7$ .

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

Schritt 5.1.37

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.38

Potenziere $2$ mit $3$ .

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

Schritt 5.1.39

Kürze den gemeinsamen Faktor von $8$ .

Tippen, um mehr Schritte zu sehen ...

Schritt 5.1.39.1

Faktorisiere $8$ aus $56$ heraus.

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

Schritt 5.1.39.2

Kürze den gemeinsamen Faktor.

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

Schritt 5.1.39.3

Forme den Ausdruck um.

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

Schritt 5.1.40

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

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Schritt 5.1.40.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.40.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.41

Vereinfache den Zähler.

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Schritt 5.1.41.1

Faktorisiere $i^{4}$ aus.

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

Schritt 5.1.41.2

Schreibe $i^{4}$ als $1$ um.

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Schritt 5.1.41.2.1

Schreibe $i^{4}$ als ${(i^{2})}^{2}$ um.

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

Schritt 5.1.41.2.2

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.41.2.3

Potenziere $- 1$ mit $2$ .

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

Schritt 5.1.41.3

Mutltipliziere $i$ mit $1$ .

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

Schritt 5.1.42

Potenziere $2$ mit $5$ .

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

Schritt 5.1.43

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

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Schritt 5.1.43.1

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

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

Schritt 5.1.43.2

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

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

Schritt 5.1.44

Bringe $7$ auf die linke Seite von $i \sin^{5} (\frac{π}{16})$ .

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

Schritt 5.1.45

Wende die Produktregel auf $\frac{\cos (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.46

Potenziere $2$ mit $2$ .

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

Schritt 5.1.47

Kürze den gemeinsamen Faktor von $4$ .

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Schritt 5.1.47.1

Faktorisiere $4$ aus $28$ heraus.

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

Schritt 5.1.47.2

Kürze den gemeinsamen Faktor.

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

Schritt 5.1.47.3

Forme den Ausdruck um.

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

Schritt 5.1.48

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

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Schritt 5.1.48.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.48.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.49

Vereinfache den Zähler.

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Schritt 5.1.49.1

Faktorisiere $i^{4}$ aus.

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

Schritt 5.1.49.2

Schreibe $i^{4}$ als $1$ um.

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Schritt 5.1.49.2.1

Schreibe $i^{4}$ als ${(i^{2})}^{2}$ um.

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

Schritt 5.1.49.2.2

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.49.2.3

Potenziere $- 1$ mit $2$ .

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

Schritt 5.1.49.3

Mutltipliziere $i^{2}$ mit $1$ .

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

Schritt 5.1.49.4

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.50

Potenziere $2$ mit $6$ .

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

Schritt 5.1.51

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 5.1.52

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

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Schritt 5.1.52.1

Mutltipliziere $- 1$ mit $7$ .

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

Schritt 5.1.52.2

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

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

Schritt 5.1.52.3

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

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

Schritt 5.1.53

Bringe $- 7$ auf die linke Seite von $\sin^{6} (\frac{π}{16})$ .

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

Schritt 5.1.54

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 5.1.55

Kürze den gemeinsamen Faktor von $2$ .

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Schritt 5.1.55.1

Faktorisiere $2$ aus $8$ heraus.

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

Schritt 5.1.55.2

Kürze den gemeinsamen Faktor.

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

Schritt 5.1.55.3

Forme den Ausdruck um.

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

Schritt 5.1.56

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

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Schritt 5.1.56.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.56.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.57

Vereinfache den Zähler.

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Schritt 5.1.57.1

Schreibe $i^{7}$ als $i^{4} (i^{2} \cdot i)$ um.

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Schritt 5.1.57.1.1

Faktorisiere $i^{4}$ aus.

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

Schritt 5.1.57.1.2

Faktorisiere $i^{2}$ aus.

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

Schritt 5.1.57.2

Schreibe $i^{4}$ als $1$ um.

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Schritt 5.1.57.2.1

Schreibe $i^{4}$ als ${(i^{2})}^{2}$ um.

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

Schritt 5.1.57.2.2

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.57.2.3

Potenziere $- 1$ mit $2$ .

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

Schritt 5.1.57.3

Mutltipliziere $i^{2} \cdot i$ mit $1$ .

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

Schritt 5.1.57.4

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.57.5

Schreibe $- 1 i$ als $- i$ um.

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

Schritt 5.1.58

Potenziere $2$ mit $7$ .

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

Schritt 5.1.59

Kürze den gemeinsamen Faktor von $4$ .

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Schritt 5.1.59.1

Faktorisiere $4$ aus $4 \cos (\frac{π}{16})$ heraus.

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

Schritt 5.1.59.2

Faktorisiere $4$ aus $128$ heraus.

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

Schritt 5.1.59.3

Kürze den gemeinsamen Faktor.

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

Schritt 5.1.59.4

Forme den Ausdruck um.

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

Schritt 5.1.60

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

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

Schritt 5.1.61

Faktorisiere das negative Vorzeichen heraus.

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

Schritt 5.1.62

Ziehe das Minuszeichen vor den Bruch.

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

Schritt 5.1.63

Wende die Exponentenregel ${(a b)}^{n} = a^{n} b^{n}$ an, um den Exponenten zu verteilen.

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Schritt 5.1.63.1

Wende die Produktregel auf $\frac{i \sin (\frac{π}{16})}{2}$ an.

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

Schritt 5.1.63.2

Wende die Produktregel auf $i \sin (\frac{π}{16})$ an.

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

Schritt 5.1.64

Vereinfache den Zähler.

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Schritt 5.1.64.1

Schreibe $i^{8}$ als ${(i^{4})}^{2}$ um.

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

Schritt 5.1.64.2

Schreibe $i^{4}$ als $1$ um.

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Schritt 5.1.64.2.1

Schreibe $i^{4}$ als ${(i^{2})}^{2}$ um.

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

Schritt 5.1.64.2.2

Schreibe $i^{2}$ als $- 1$ um.

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

Schritt 5.1.64.2.3

Potenziere $- 1$ mit $2$ .

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

Schritt 5.1.64.3

Eins zu einer beliebigen Potenz erhoben ergibt eins.

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

Schritt 5.1.64.4

Mutltipliziere $\sin^{8} (\frac{π}{16})$ mit $1$ .

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

Schritt 5.1.65

Potenziere $2$ mit $8$ .

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

Schritt 5.2

Vereinfache durch Vertauschen.

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Schritt 5.2.1

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

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

Schritt 5.2.2

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

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

Schritt 5.2.3

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

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

Schritt 5.2.4

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

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

Schritt 5.2.5

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

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