Question

The value of (1 + cos $\frac{\pi }{8}$)(1 + cos $\frac{3\pi }{8}$)(1 + cos $\frac{5\pi }{8}$)(1 + cos $\frac{7\pi }{8}$) is equal to

• A. 1/4
• B. 1/6
• C. 1/8
• D. 1/2

Answer 1

The correct option is C

1/8

cos $\frac{5\pi }{8}$ = cos( $\frac{\pi }{2}$ + $\frac{\pi }{8}$) = -sin $\frac{\pi }{8}$

cos $\frac{3\pi }{8}$ = cos( $\frac{\pi }{2}$ - $\frac{\pi }{8}$) = sin $\frac{\pi }{8}$

cos $\frac{7\pi }{8}$ = cos($\pi$ - $\frac{\pi }{8}$) = -cos $\frac{\pi }{8}$

⇒ (1 + cos $\frac{\pi }{8}$)(1 - cos $\frac{\pi }{8}$)(1 + sin $\frac{\pi }{8}$)(1 - sin $\frac{\pi }{8}$)

⇒ (1 - $co{s}^2$ $\frac{\pi }{8}$)(1 - $si{n}^2$ $\frac{\pi }{8}$) = $si{n}^2$ $\frac{\pi }{8}$$co{s}^2$ $\frac{\pi }{8}$

$\frac{1}{4}$$(sin\frac{\pi }{4}{)}^2$ = $\frac{1}{4}$. $\frac{1}{2}$ = $\frac{1}{8}$