JEE Main · 2021 · 22 Jul 2021 · Shift 2 · Trigonometric Equations

For every $t\in [-1,1]$, $t^7\le t^2$, with equality only at $t=0$ and $t=1$. Hence

and equality forces each of $\sin x$ and $\cos x$ to be $0$ or $1$. The only admissible pairs are $(\sin x,\cos x)=(0,1)$ and $(1,0)$. In $[0,4\pi ]$ the first occurs at $x=0,2\pi ,4\pi$ and the second at $x=\frac{\pi }{2},\frac{5\pi }{2}$. The equation has $5$ solutions.