JEE Main 2020 (Online) 9th January Morning ShiftQuestion 1
The number of distinct solutions of the equation, $\log_{\frac12}|\sin x|=2-\log_{\frac12}|\cos x|$ in the interval $[0,2\pi]$, is ________.
Trigonometric Ratio & Identities
Answer$8$
Solution
Concept. Combine the logarithms; the equation becomes $|\sin2x|=\frac12$. Formulas used. (1) $\log_ap+\log_aq=\log_a(pq)$; $\sin x\cos x=\frac12\sin2x$Step 1 - Combine. $\log_{1/2}|\sin x\cos x|=2$, so $|\sin x\cos x|=\frac14$ and $|\sin2x|=\frac12$. Step 2 - Count. $2x\in[0,4\pi]$; $|\sin t|=\frac12$ has $4$ solutions in each $2\pi$, so $8$ in all. At none of them is $\sin x$ or $\cos x$ zero. Answer: 8