Trigonometric Ratios and Identities
Product of Cosines
JEE Main 2023
Grade 11
Question:
The value of $36(4 \cos^2 9^\circ - 1)(4 \cos^2 27^\circ - 1)(4 \cos^2 81^\circ - 1)(4 \cos^2 243^\circ - 1)$ is:
(1) $12$
(2) $18$
(3) $36$
(4) $54$
Step-by-Step Solution
Key Concept: Derive $4 \cos^2 \theta - 1 = \sin 3\theta / \sin \theta$ from $\sin 3\theta = \sin \theta(4 \cos^2 \theta - 1)$. Write the product as $\frac{\sin 27^\circ}{\sin 9^\circ} \cdot \frac{\sin 81^\circ}{\sin 27^\circ} \dots$ and telescope.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)