Trigonometric Ratios and Identities
Product of Cosines
JEE Main 2023
Grade 11
Question:
The value of $96 \cos \frac{\pi}{33} \cos \frac{2\pi}{33} \cos \frac{4\pi}{33} \cos \frac{8\pi}{33} \cos \frac{16\pi}{33}$ is:
(1) $3$
(2) $6$
(3) $2$
(4) $\sqrt{3}$
Step-by-Step Solution
Key Concept: Use the identity $\prod_{k=0}^{n-1} \cos(2^k \theta) = \frac{\sin(2^n \theta)}{2^n \sin \theta}$. Choose $\theta = \pi/33$ and $n = 5$. Note that $\sin(32\pi/33) = \sin(\pi/33)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)