Đại số Ví dụ

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

Bước 1

Áp dụng thuộc tính phân phối.

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

Bước 2

Kết hợp $\frac{1}{2}$ và $\cos (\frac{π}{16})$ .

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

Bước 3

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

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Bước 3.1

Kết hợp $i$ và $\frac{1}{2}$ .

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

Bước 3.2

Kết hợp $\frac{i}{2}$ và $\sin (\frac{π}{16})$ .

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

Bước 4

Sử dụng định lý nhị thức.

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

Bước 5

Rút gọn các số hạng.

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Bước 5.1

Rút gọn mỗi số hạng.

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Bước 5.1.1

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.2

Nâng $2$ lên lũy thừa $8$ .

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

Bước 5.1.3

Triệt tiêu thừa số chung $2$ .

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Bước 5.1.3.1

Đưa $2$ ra ngoài $8 {(\frac{\cos (\frac{π}{16})}{2})}^{7}$ .

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

Bước 5.1.3.2

Triệt tiêu thừa số chung.

Bước 5.1.3.3

Viết lại biểu thức.

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

Bước 5.1.4

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.5

Nâng $2$ lên lũy thừa $7$ .

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

Bước 5.1.6

Triệt tiêu thừa số chung $4$ .

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Bước 5.1.6.1

Đưa $4$ ra ngoài $128$ .

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

Bước 5.1.6.2

Triệt tiêu thừa số chung.

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

Bước 5.1.6.3

Viết lại biểu thức.

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

Bước 5.1.7

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

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Bước 5.1.7.1

Kết hợp $i$ và $\frac{\cos^{7} (\frac{π}{16})}{32}$ .

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

Bước 5.1.7.2

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

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

Bước 5.1.8

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.9

Nâng $2$ lên lũy thừa $6$ .

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

Bước 5.1.10

Triệt tiêu thừa số chung $4$ .

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Bước 5.1.10.1

Đưa $4$ ra ngoài $28$ .

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

Bước 5.1.10.2

Đưa $4$ ra ngoài $64$ .

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

Bước 5.1.10.3

Triệt tiêu thừa số chung.

Bước 5.1.10.4

Viết lại biểu thức.

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

Bước 5.1.11

Kết hợp $7$ và $\frac{\cos^{6} (\frac{π}{16})}{16}$ .

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

Bước 5.1.12

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.12.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.12.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.13

Kết hợp.

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

Bước 5.1.14

Rút gọn tử số.

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Bước 5.1.14.1

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.14.2

Kết hợp các số mũ.

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Bước 5.1.14.2.1

Đưa dấu âm ra ngoài.

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

Bước 5.1.14.2.2

Nhân $7$ với $- 1$ .

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

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

Bước 5.1.15

Rút gọn mẫu số.

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Bước 5.1.15.1

Viết lại $16$ ở dạng $2^{4}$ .

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

Bước 5.1.15.2

Sử dụng quy tắc lũy thừa $a^{m} a^{n} = a^{m + n}$ để kết hợp các số mũ.

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

Bước 5.1.15.3

Cộng $4$ và $2$ .

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

Bước 5.1.16

Nâng $2$ lên lũy thừa $6$ .

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

Bước 5.1.17

Di chuyển dấu trừ ra phía trước của phân số.

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

Bước 5.1.18

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.19

Nâng $2$ lên lũy thừa $5$ .

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

Bước 5.1.20

Triệt tiêu thừa số chung $8$ .

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Bước 5.1.20.1

Đưa $8$ ra ngoài $56$ .

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

Bước 5.1.20.2

Đưa $8$ ra ngoài $32$ .

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

Bước 5.1.20.3

Triệt tiêu thừa số chung.

Bước 5.1.20.4

Viết lại biểu thức.

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

Bước 5.1.21

Kết hợp $7$ và $\frac{\cos^{5} (\frac{π}{16})}{4}$ .

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

Bước 5.1.22

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.22.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.22.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.23

Kết hợp.

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

Bước 5.1.24

Rút gọn tử số.

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Bước 5.1.24.1

Đưa $i^{2}$ ra ngoài.

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

Bước 5.1.24.2

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.24.3

Viết lại $- 1 i$ ở dạng $- i$ .

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

Bước 5.1.24.4

Kết hợp các số mũ.

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Bước 5.1.24.4.1

Đưa dấu âm ra ngoài.

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

Bước 5.1.24.4.2

Nhân $7$ với $- 1$ .

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

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

Bước 5.1.25

Rút gọn mẫu số.

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Bước 5.1.25.1

Viết lại $4$ ở dạng $2^{2}$ .

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

Bước 5.1.25.2

Sử dụng quy tắc lũy thừa $a^{m} a^{n} = a^{m + n}$ để kết hợp các số mũ.

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

Bước 5.1.25.3

Cộng $2$ và $3$ .

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

Bước 5.1.26

Nâng $2$ lên lũy thừa $5$ .

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

Bước 5.1.27

Di chuyển dấu trừ ra phía trước của phân số.

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

Bước 5.1.28

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.29

Nâng $2$ lên lũy thừa $4$ .

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

Bước 5.1.30

Triệt tiêu thừa số chung $2$ .

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Bước 5.1.30.1

Đưa $2$ ra ngoài $70$ .

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

Bước 5.1.30.2

Đưa $2$ ra ngoài $16$ .

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

Bước 5.1.30.3

Triệt tiêu thừa số chung.

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

Bước 5.1.30.4

Viết lại biểu thức.

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

Bước 5.1.31

Kết hợp $35$ và $\frac{\cos^{4} (\frac{π}{16})}{8}$ .

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

Bước 5.1.32

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.32.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.32.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.33

Kết hợp.

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

Bước 5.1.34

Rút gọn tử số.

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Bước 5.1.34.1

Viết lại $i^{4}$ ở dạng $1$ .

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Bước 5.1.34.1.1

Viết lại $i^{4}$ ở dạng ${(i^{2})}^{2}$ .

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

Bước 5.1.34.1.2

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.34.1.3

Nâng $- 1$ lên lũy thừa $2$ .

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

Bước 5.1.34.2

Nhân $35$ với $1$ .

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

Bước 5.1.35

Rút gọn mẫu số.

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Bước 5.1.35.1

Viết lại $8$ ở dạng $2^{3}$ .

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

Bước 5.1.35.2

Sử dụng quy tắc lũy thừa $a^{m} a^{n} = a^{m + n}$ để kết hợp các số mũ.

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

Bước 5.1.35.3

Cộng $3$ và $4$ .

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

Bước 5.1.36

Nâng $2$ lên lũy thừa $7$ .

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

Bước 5.1.37

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.38

Nâng $2$ lên lũy thừa $3$ .

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

Bước 5.1.39

Triệt tiêu thừa số chung $8$ .

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Bước 5.1.39.1

Đưa $8$ ra ngoài $56$ .

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

Bước 5.1.39.2

Triệt tiêu thừa số chung.

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

Bước 5.1.39.3

Viết lại biểu thức.

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

Bước 5.1.40

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.40.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.40.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.41

Rút gọn tử số.

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Bước 5.1.41.1

Đưa $i^{4}$ ra ngoài.

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

Bước 5.1.41.2

Viết lại $i^{4}$ ở dạng $1$ .

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Bước 5.1.41.2.1

Viết lại $i^{4}$ ở dạng ${(i^{2})}^{2}$ .

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

Bước 5.1.41.2.2

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.41.2.3

Nâng $- 1$ lên lũy thừa $2$ .

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

Bước 5.1.41.3

Nhân $i$ với $1$ .

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

Bước 5.1.42

Nâng $2$ lên lũy thừa $5$ .

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

Bước 5.1.43

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

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Bước 5.1.43.1

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

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

Bước 5.1.43.2

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

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

Bước 5.1.44

Di chuyển $7$ sang phía bên trái của $i \sin^{5} (\frac{π}{16})$ .

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

Bước 5.1.45

Áp dụng quy tắc tích số cho $\frac{\cos (\frac{π}{16})}{2}$ .

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

Bước 5.1.46

Nâng $2$ lên lũy thừa $2$ .

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

Bước 5.1.47

Triệt tiêu thừa số chung $4$ .

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Bước 5.1.47.1

Đưa $4$ ra ngoài $28$ .

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

Bước 5.1.47.2

Triệt tiêu thừa số chung.

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

Bước 5.1.47.3

Viết lại biểu thức.

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

Bước 5.1.48

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.48.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.48.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.49

Rút gọn tử số.

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Bước 5.1.49.1

Đưa $i^{4}$ ra ngoài.

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

Bước 5.1.49.2

Viết lại $i^{4}$ ở dạng $1$ .

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Bước 5.1.49.2.1

Viết lại $i^{4}$ ở dạng ${(i^{2})}^{2}$ .

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

Bước 5.1.49.2.2

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.49.2.3

Nâng $- 1$ lên lũy thừa $2$ .

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

Bước 5.1.49.3

Nhân $i^{2}$ với $1$ .

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

Bước 5.1.49.4

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.50

Nâng $2$ lên lũy thừa $6$ .

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

Bước 5.1.51

Di chuyển dấu trừ ra phía trước của phân số.

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

Bước 5.1.52

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

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Bước 5.1.52.1

Nhân $- 1$ với $7$ .

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

Bước 5.1.52.2

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

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

Bước 5.1.52.3

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

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

Bước 5.1.53

Di chuyển $- 7$ sang phía bên trái của $\sin^{6} (\frac{π}{16})$ .

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

Bước 5.1.54

Di chuyển dấu trừ ra phía trước của phân số.

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

Bước 5.1.55

Triệt tiêu thừa số chung $2$ .

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Bước 5.1.55.1

Đưa $2$ ra ngoài $8$ .

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

Bước 5.1.55.2

Triệt tiêu thừa số chung.

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

Bước 5.1.55.3

Viết lại biểu thức.

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

Bước 5.1.56

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.56.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.56.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.57

Rút gọn tử số.

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Bước 5.1.57.1

Viết lại $i^{7}$ ở dạng $i^{4} (i^{2} \cdot i)$ .

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Bước 5.1.57.1.1

Đưa $i^{4}$ ra ngoài.

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

Bước 5.1.57.1.2

Đưa $i^{2}$ ra ngoài.

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

Bước 5.1.57.2

Viết lại $i^{4}$ ở dạng $1$ .

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Bước 5.1.57.2.1

Viết lại $i^{4}$ ở dạng ${(i^{2})}^{2}$ .

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

Bước 5.1.57.2.2

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.57.2.3

Nâng $- 1$ lên lũy thừa $2$ .

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

Bước 5.1.57.3

Nhân $i^{2} \cdot i$ với $1$ .

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

Bước 5.1.57.4

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.57.5

Viết lại $- 1 i$ ở dạng $- i$ .

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

Bước 5.1.58

Nâng $2$ lên lũy thừa $7$ .

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

Bước 5.1.59

Triệt tiêu thừa số chung $4$ .

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Bước 5.1.59.1

Đưa $4$ ra ngoài $4 \cos (\frac{π}{16})$ .

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

Bước 5.1.59.2

Đưa $4$ ra ngoài $128$ .

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

Bước 5.1.59.3

Triệt tiêu thừa số chung.

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

Bước 5.1.59.4

Viết lại biểu thức.

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

Bước 5.1.60

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

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

Bước 5.1.61

Đưa dấu âm ra ngoài.

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

Bước 5.1.62

Di chuyển dấu trừ ra phía trước của phân số.

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

Bước 5.1.63

Sử dụng quy tắc lũy thừa ${(a b)}^{n} = a^{n} b^{n}$ để phân phối các số mũ.

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Bước 5.1.63.1

Áp dụng quy tắc tích số cho $\frac{i \sin (\frac{π}{16})}{2}$ .

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

Bước 5.1.63.2

Áp dụng quy tắc tích số cho $i \sin (\frac{π}{16})$ .

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

Bước 5.1.64

Rút gọn tử số.

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Bước 5.1.64.1

Viết lại $i^{8}$ ở dạng ${(i^{4})}^{2}$ .

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

Bước 5.1.64.2

Viết lại $i^{4}$ ở dạng $1$ .

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Bước 5.1.64.2.1

Viết lại $i^{4}$ ở dạng ${(i^{2})}^{2}$ .

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

Bước 5.1.64.2.2

Viết lại $i^{2}$ ở dạng $- 1$ .

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

Bước 5.1.64.2.3

Nâng $- 1$ lên lũy thừa $2$ .

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

Bước 5.1.64.3

Một mũ bất kỳ số nào là một.

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

Bước 5.1.64.4

Nhân $\sin^{8} (\frac{π}{16})$ với $1$ .

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

Bước 5.1.65

Nâng $2$ lên lũy thừa $8$ .

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

Bước 5.2

Rút gọn bằng cách giao hoán.

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Bước 5.2.1

Sắp xếp lại các thừa số trong $\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32} - \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64} - \frac{7 \cos^{5} (\frac{π}{16}) i \sin^{3} (\frac{π}{16})}{32} + \frac{35 \cos^{4} (\frac{π}{16}) \sin^{4} (\frac{π}{16})}{128} + \frac{7 i \sin^{5} (\frac{π}{16}) \cos^{3} (\frac{π}{16})}{32} - \frac{7 \sin^{6} (\frac{π}{16}) \cos^{2} (\frac{π}{16})}{64} - \frac{\cos (\frac{π}{16}) i \sin^{7} (\frac{π}{16})}{32} + \frac{\sin^{8} (\frac{π}{16})}{256}$ .

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

Bước 5.2.2

Sắp xếp lại $\frac{\cos^{8} (\frac{π}{16})}{256} + \frac{i \cos^{7} (\frac{π}{16}) \sin (\frac{π}{16})}{32}$ và $- \frac{7 \cos^{6} (\frac{π}{16}) \sin^{2} (\frac{π}{16})}{64}$ .

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

Bước 5.2.3

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

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

Bước 5.2.4

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

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

Bước 5.2.5

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

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