代数示例

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

解题步骤 1

运用分配律。

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

解题步骤 2

组合 $\frac{1}{2}$ 和 $\cos (\frac{π}{16})$ 。

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

解题步骤 3

乘以 $\frac{1}{2} (i \sin (\frac{π}{16}))$ 。

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解题步骤 3.1

组合 $i$ 和 $\frac{1}{2}$ 。

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

解题步骤 3.2

组合 $\frac{i}{2}$ 和 $\sin (\frac{π}{16})$ 。

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

解题步骤 4

使用二项式定理。

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

解题步骤 5

化简项。

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解题步骤 5.1

化简每一项。

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解题步骤 5.1.1

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.2

对 $2$ 进行 $8$ 次方运算。

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

解题步骤 5.1.3

约去 $2$ 的公因数。

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解题步骤 5.1.3.1

从 $8 {(\frac{\cos (\frac{π}{16})}{2})}^{7}$ 中分解出因数 $2$ 。

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

解题步骤 5.1.3.2

约去公因数。

解题步骤 5.1.3.3

重写表达式。

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

解题步骤 5.1.4

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.5

对 $2$ 进行 $7$ 次方运算。

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

解题步骤 5.1.6

约去 $4$ 的公因数。

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解题步骤 5.1.6.1

从 $128$ 中分解出因数 $4$ 。

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

解题步骤 5.1.6.2

约去公因数。

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

解题步骤 5.1.6.3

重写表达式。

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

解题步骤 5.1.7

乘以 $\frac{\cos^{7} (\frac{π}{16})}{32} (i \sin (\frac{π}{16}))$ 。

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解题步骤 5.1.7.1

组合 $i$ 和 $\frac{\cos^{7} (\frac{π}{16})}{32}$ 。

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

解题步骤 5.1.7.2

组合 $\frac{i \cos^{7} (\frac{π}{16})}{32}$ 和 $\sin (\frac{π}{16})$ 。

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

解题步骤 5.1.8

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.9

对 $2$ 进行 $6$ 次方运算。

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

解题步骤 5.1.10

约去 $4$ 的公因数。

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解题步骤 5.1.10.1

从 $28$ 中分解出因数 $4$ 。

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

解题步骤 5.1.10.2

从 $64$ 中分解出因数 $4$ 。

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

解题步骤 5.1.10.3

约去公因数。

解题步骤 5.1.10.4

重写表达式。

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

解题步骤 5.1.11

组合 $7$ 和 $\frac{\cos^{6} (\frac{π}{16})}{16}$ 。

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

解题步骤 5.1.12

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.12.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.12.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.13

合并。

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

解题步骤 5.1.14

化简分子。

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解题步骤 5.1.14.1

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.14.2

合并指数。

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解题步骤 5.1.14.2.1

提取负因数。

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

解题步骤 5.1.14.2.2

将 $7$ 乘以 $- 1$ 。

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

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

解题步骤 5.1.15

化简分母。

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解题步骤 5.1.15.1

将 $16$ 重写为 $2^{4}$ 。

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

解题步骤 5.1.15.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 5.1.15.3

将 $4$ 和 $2$ 相加。

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

解题步骤 5.1.16

对 $2$ 进行 $6$ 次方运算。

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

解题步骤 5.1.17

将负号移到分数的前面。

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

解题步骤 5.1.18

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.19

对 $2$ 进行 $5$ 次方运算。

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

解题步骤 5.1.20

约去 $8$ 的公因数。

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解题步骤 5.1.20.1

从 $56$ 中分解出因数 $8$ 。

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

解题步骤 5.1.20.2

从 $32$ 中分解出因数 $8$ 。

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

解题步骤 5.1.20.3

约去公因数。

解题步骤 5.1.20.4

重写表达式。

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

解题步骤 5.1.21

组合 $7$ 和 $\frac{\cos^{5} (\frac{π}{16})}{4}$ 。

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

解题步骤 5.1.22

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.22.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.22.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.23

合并。

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

解题步骤 5.1.24

化简分子。

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解题步骤 5.1.24.1

因式分解出 $i^{2}$ 。

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

解题步骤 5.1.24.2

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.24.3

将 $- 1 i$ 重写为 $- i$ 。

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

解题步骤 5.1.24.4

合并指数。

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解题步骤 5.1.24.4.1

提取负因数。

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

解题步骤 5.1.24.4.2

将 $7$ 乘以 $- 1$ 。

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

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

解题步骤 5.1.25

化简分母。

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解题步骤 5.1.25.1

将 $4$ 重写为 $2^{2}$ 。

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

解题步骤 5.1.25.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 5.1.25.3

将 $2$ 和 $3$ 相加。

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

解题步骤 5.1.26

对 $2$ 进行 $5$ 次方运算。

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

解题步骤 5.1.27

将负号移到分数的前面。

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

解题步骤 5.1.28

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.29

对 $2$ 进行 $4$ 次方运算。

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

解题步骤 5.1.30

约去 $2$ 的公因数。

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解题步骤 5.1.30.1

从 $70$ 中分解出因数 $2$ 。

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

解题步骤 5.1.30.2

从 $16$ 中分解出因数 $2$ 。

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

解题步骤 5.1.30.3

约去公因数。

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

解题步骤 5.1.30.4

重写表达式。

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

解题步骤 5.1.31

组合 $35$ 和 $\frac{\cos^{4} (\frac{π}{16})}{8}$ 。

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

解题步骤 5.1.32

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.32.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.32.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.33

合并。

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

解题步骤 5.1.34

化简分子。

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解题步骤 5.1.34.1

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 5.1.34.1.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

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

解题步骤 5.1.34.1.2

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.34.1.3

对 $- 1$ 进行 $2$ 次方运算。

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

解题步骤 5.1.34.2

将 $35$ 乘以 $1$ 。

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

解题步骤 5.1.35

化简分母。

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解题步骤 5.1.35.1

将 $8$ 重写为 $2^{3}$ 。

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

解题步骤 5.1.35.2

使用幂法则 $a^{m} a^{n} = a^{m + n}$ 合并指数。

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

解题步骤 5.1.35.3

将 $3$ 和 $4$ 相加。

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

解题步骤 5.1.36

对 $2$ 进行 $7$ 次方运算。

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

解题步骤 5.1.37

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.38

对 $2$ 进行 $3$ 次方运算。

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

解题步骤 5.1.39

约去 $8$ 的公因数。

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解题步骤 5.1.39.1

从 $56$ 中分解出因数 $8$ 。

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

解题步骤 5.1.39.2

约去公因数。

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

解题步骤 5.1.39.3

重写表达式。

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

解题步骤 5.1.40

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.40.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.40.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.41

化简分子。

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解题步骤 5.1.41.1

因式分解出 $i^{4}$ 。

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

解题步骤 5.1.41.2

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 5.1.41.2.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

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

解题步骤 5.1.41.2.2

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.41.2.3

对 $- 1$ 进行 $2$ 次方运算。

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

解题步骤 5.1.41.3

将 $i$ 乘以 $1$ 。

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

解题步骤 5.1.42

对 $2$ 进行 $5$ 次方运算。

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

解题步骤 5.1.43

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

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解题步骤 5.1.43.1

组合 $\frac{i \sin^{5} (\frac{π}{16})}{32}$ 和 $7$ 。

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

解题步骤 5.1.43.2

组合 $\frac{i \sin^{5} (\frac{π}{16}) \cdot 7}{32}$ 和 $\cos^{3} (\frac{π}{16})$ 。

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

解题步骤 5.1.44

将 $7$ 移到 $i \sin^{5} (\frac{π}{16})$ 的左侧。

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

解题步骤 5.1.45

对 $\frac{\cos (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.46

对 $2$ 进行 $2$ 次方运算。

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

解题步骤 5.1.47

约去 $4$ 的公因数。

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解题步骤 5.1.47.1

从 $28$ 中分解出因数 $4$ 。

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

解题步骤 5.1.47.2

约去公因数。

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

解题步骤 5.1.47.3

重写表达式。

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

解题步骤 5.1.48

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.48.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.48.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.49

化简分子。

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解题步骤 5.1.49.1

因式分解出 $i^{4}$ 。

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

解题步骤 5.1.49.2

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 5.1.49.2.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

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

解题步骤 5.1.49.2.2

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.49.2.3

对 $- 1$ 进行 $2$ 次方运算。

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

解题步骤 5.1.49.3

将 $i^{2}$ 乘以 $1$ 。

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

解题步骤 5.1.49.4

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.50

对 $2$ 进行 $6$ 次方运算。

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

解题步骤 5.1.51

将负号移到分数的前面。

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

解题步骤 5.1.52

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

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解题步骤 5.1.52.1

将 $- 1$ 乘以 $7$ 。

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

解题步骤 5.1.52.2

组合 $\frac{\sin^{6} (\frac{π}{16})}{64}$ 和 $- 7$ 。

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

解题步骤 5.1.52.3

组合 $\frac{\sin^{6} (\frac{π}{16}) \cdot - 7}{64}$ 和 $\cos^{2} (\frac{π}{16})$ 。

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

解题步骤 5.1.53

将 $- 7$ 移到 $\sin^{6} (\frac{π}{16})$ 的左侧。

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

解题步骤 5.1.54

将负号移到分数的前面。

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

解题步骤 5.1.55

约去 $2$ 的公因数。

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解题步骤 5.1.55.1

从 $8$ 中分解出因数 $2$ 。

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

解题步骤 5.1.55.2

约去公因数。

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

解题步骤 5.1.55.3

重写表达式。

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

解题步骤 5.1.56

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.56.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.56.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.57

化简分子。

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解题步骤 5.1.57.1

将 $i^{7}$ 重写为 $i^{4} (i^{2} \cdot i)$ 。

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解题步骤 5.1.57.1.1

因式分解出 $i^{4}$ 。

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

解题步骤 5.1.57.1.2

因式分解出 $i^{2}$ 。

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

解题步骤 5.1.57.2

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 5.1.57.2.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

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

解题步骤 5.1.57.2.2

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.57.2.3

对 $- 1$ 进行 $2$ 次方运算。

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

解题步骤 5.1.57.3

将 $i^{2} \cdot i$ 乘以 $1$ 。

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

解题步骤 5.1.57.4

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.57.5

将 $- 1 i$ 重写为 $- i$ 。

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

解题步骤 5.1.58

对 $2$ 进行 $7$ 次方运算。

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

解题步骤 5.1.59

约去 $4$ 的公因数。

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解题步骤 5.1.59.1

从 $4 \cos (\frac{π}{16})$ 中分解出因数 $4$ 。

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

解题步骤 5.1.59.2

从 $128$ 中分解出因数 $4$ 。

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

解题步骤 5.1.59.3

约去公因数。

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

解题步骤 5.1.59.4

重写表达式。

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

解题步骤 5.1.60

组合 $\cos (\frac{π}{16})$ 和 $\frac{- i \sin^{7} (\frac{π}{16})}{32}$ 。

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

解题步骤 5.1.61

提取负因数。

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

解题步骤 5.1.62

将负号移到分数的前面。

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

解题步骤 5.1.63

使用幂法则 ${(a b)}^{n} = a^{n} b^{n}$ 分解指数。

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解题步骤 5.1.63.1

对 $\frac{i \sin (\frac{π}{16})}{2}$ 运用乘积法则。

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

解题步骤 5.1.63.2

对 $i \sin (\frac{π}{16})$ 运用乘积法则。

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

解题步骤 5.1.64

化简分子。

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解题步骤 5.1.64.1

将 $i^{8}$ 重写为 ${(i^{4})}^{2}$ 。

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

解题步骤 5.1.64.2

将 $i^{4}$ 重写为 $1$ 。

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解题步骤 5.1.64.2.1

将 $i^{4}$ 重写为 ${(i^{2})}^{2}$ 。

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

解题步骤 5.1.64.2.2

将 $i^{2}$ 重写为 $- 1$ 。

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

解题步骤 5.1.64.2.3

对 $- 1$ 进行 $2$ 次方运算。

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

解题步骤 5.1.64.3

一的任意次幂都为一。

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

解题步骤 5.1.64.4

将 $\sin^{8} (\frac{π}{16})$ 乘以 $1$ 。

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

解题步骤 5.1.65

对 $2$ 进行 $8$ 次方运算。

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

解题步骤 5.2

通过交换进行化简。

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解题步骤 5.2.1

将 $\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$ 中的因式重新排序。

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

解题步骤 5.2.2

将 $\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32}$ 和 $- \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64}$ 重新排序。

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

解题步骤 5.2.3

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

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

解题步骤 5.2.4

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

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

解题步骤 5.2.5

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

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

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