Question

The number of values of $\theta$ in the interval $(-\frac{\pi }{2},\frac{\pi }{2})$ such that $\tan \theta =\cot 5\theta$ as well as $\cos 2\theta =\sin 4\theta$ is

Answer 1

$\tan \theta =\cot 5\theta \Rightarrow \tan \theta =\tan (\frac{\pi }{2}-5\theta )\Rightarrow \theta =n\pi +\frac{\pi }{2}-5\theta \Rightarrow \theta =\frac{n\pi }{6}+\frac{\pi }{12},n\in Z$
$\therefore \theta =\{\pm \frac{\pi }{12},\pm \frac{\pi }{4},\pm \frac{5\pi }{12}\}\cdots \left(i\right)$
Now $\cos 2\theta =\sin 4\theta$
$\cos 2\theta =\cos (\frac{\pi }{2}-4\theta )\Rightarrow 2\theta =2n\pi \pm (\frac{\pi }{2}-4\theta )\Rightarrow \theta =\frac{(4n+1)\pi }{12},-\frac{(4n-1)\pi }{4},n\in Z\therefore \theta =\{-\frac{\pi }{4},\frac{\pi }{12},\frac{\pi }{4},\frac{5\pi }{12}\}\cdots \left(ii\right)$
from $\left(i\right)$ and $\left(ii\right)$
$\theta =\{-\frac{\pi }{4},\frac{\pi }{12},\frac{\pi }{4},\frac{5\pi }{12}\}$