Question

Evaluate \[ (4 \cos^2 \frac{\pi}{20} - 1)(4 \cos^2 \frac{3\pi}{20} - 1)(4 \cos^2 \frac{5\pi}{20} - 1)(4 \cos^2 \frac{7\pi}{20} - 1)(4 \cos^2 \frac{9\pi}{20} - 1). \]

• A. 1
• B. $\frac{1}{2}$
• C. 2
• D. 3

Hint:

Use double-angle formulas and known cosine product identities for special angle products.

Answer 1

The Correct Option is D

Use identity $4 \cos^2 \theta - 1 = 2 \cos 2\theta$. Then product becomes:
\[ \prod_{k=1}^5 (4 \cos^2 \frac{(2k-1)\pi}{20} - 1) = \prod_{k=1}^5 2 \cos \frac{(2k-1) \pi}{10}. \] Extract constant factors $2^5 = 32$. The product reduces to $32 \times \prod_{k=1}^5 \cos \frac{(2k-1)\pi}{10}$. Using known product formulas for cosines of equally spaced angles, this product equals $\frac{3}{32}$. Multiply and get the result $3$.