Trigonometric Ratios and Identities
Product of Sines
JEE Main 2021
Grade 11
Question:
The value of $2 \sin \frac{\pi}{8} \sin \frac{2\pi}{8} \sin \frac{3\pi}{8} \sin \frac{5\pi}{8} \sin \frac{6\pi}{8} \sin \frac{7\pi}{8}$ is:
(1) $\frac{1}{4}$
(2) $\frac{1}{8}$
(3) $\frac{\sqrt{2}}{8}$
(4) $\frac{1}{4\sqrt{2}}$
Step-by-Step Solution
Key Concept: Use $\sin(\pi - x) = \sin x$ to pair: $\sin(7\pi/8) = \sin(\pi/8)$, $\sin(6\pi/8) = \sin(2\pi/8)$, $\sin(5\pi/8) = \sin(3\pi/8)$. The product reduces to $2 \sin^2(\pi/8) \sin^2(\pi/4) \sin^2(\3\pi/8)$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)