Question

Total number of solution of the equation $\cos 4x\cos x=\sin 4x\sin x$ where $x\in [0,\pi ]$ is

Answer 1

$\cos 4x\cos x=\sin 4x\sin x$
$\Rightarrow \cos 4x\cos x-\sin 4x\sin x=0$
$\Rightarrow \cos 5x=0$
$\Rightarrow 5x=(2n+1)\frac{\pi }{2}$
$\Rightarrow x=\left(\frac{2n+1}{10}\right)\pi$
$\because 0\le x\le \pi$
$\therefore x=\frac{\pi }{10},\frac{3\pi }{10},\frac{5\pi }{10},\frac{7\pi }{10},\frac{9\pi }{10}$

$\therefore$ Total number of solutions is $5.$