Question

Number of solution(s) of the equation $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x}=0$ in the interval $(0,\frac{\pi }{4})$ is

Answer 1

$\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x}=0$
$\frac{2\sin x\cos x}{2\cos 3x\cos x}+\frac{2\sin 3x\cos 3x}{2\cos 9x\cos 3x}+\frac{2\sin 9x\cos 9x}{2\cos 27x\cos 9x}=0$
$\frac{1}{2}(\frac{\sin (3x-x)}{\cos 3x\cos x}+\frac{\sin (9x-3x)}{\cos 9x\cos 3x}+\frac{\sin (27-9x)}{\cos 27x\cos 9x})=0$
$\frac{\sin 3x\cos x-\cos 3x\sin x}{\cos 3x\cos x}+\frac{\sin 9x\cos 3x-\cos 9x\sin 3x}{\cos 9x\cos 3x}+\frac{\sin 27x\cos 9x-\cos 27x\sin 9x}{\cos 27x\cos 9x}=0$
$\Rightarrow \tan 3x-\tan x+\tan 9x-\tan 3x+\tan 27x-\tan 9x=0$
$\Rightarrow \tan 27x-\tan x=0$
$\Rightarrow 26x=n\pi ,n\in I$
$\Rightarrow x=\frac{n\pi }{26}$
For $x\in (0,\frac{\pi }{4})$
$\Rightarrow x=0,\frac{\pi }{26},\frac{2\pi }{26},\frac{3\pi }{26},\frac{4\pi }{26},\frac{5\pi }{26},\frac{6\pi }{26}$
Hence, number of solutions$=6$