Question

If \(33θ = π\), then the value of \(cosθcos2θcos4θcos8θcos16θ\) is

• A. \(\frac{1}{16}\)
• B. \(\frac{1}{32}\)
• C. \(\frac{1}{64}\)
• D. \(\frac{-1}{32}\)

Hint:

Multiply and divide by sin(theta) and use the double angle formula repeatedly.

Answer 1

The Correct Option is B

Step 1: Idea
Five cosines with angles doubling suggest $\sin 2^5\theta / (2^5 \sin\theta)$.

Step 2: Result of telescoping
$P = \dfrac{\sin 32\theta}{32\sin\theta}$.

Step 3: Substitute
$\theta = \pi/33$ gives $32\theta = \pi - \theta$, so $\sin 32\theta = \sin\theta$ and $P = 1/32$. Option (B).

Final Answer:
Option (B). \[ \boxed{\frac{1}{32}} \]