JEE Main 2023 (Online) 10th April Morning ShiftQuestion 1
$96\cos\frac{\pi}{33}\cos\frac{2\pi}{33}\cos\frac{4\pi}{33}\cos\frac{8\pi}{33}\cos\frac{16\pi}{33}$ is equal to
Trigonometric Ratio & Identities
Solution
Concept. Multiply by $\sin\theta$ and use $2\sin x\cos x=\sin2x$ five times. Formulas used. (1) $\sin\theta\cos\theta\cos2\theta\cos4\theta\cos8\theta\cos16\theta=\dfrac{\sin32\theta}{32}$Step 1 - Apply (1). With $\theta=\frac{\pi}{33}$: $$\prod=\frac{\sin\frac{32\pi}{33}}{32\sin\frac{\pi}{33}}.$$ Step 2 - Simplify. $\sin\frac{32\pi}{33}=\sin\left(\pi-\frac{\pi}{33}\right)=\sin\frac{\pi}{33}$, so the product is $\frac1{32}$. Step 3 - Value. $96\cdot\frac{1}{32}=3$. Answer: (C)