Question

Solve $\tan 5\theta =\cot 2\theta .$

Answer 1

Simplifying $\tan 5\theta =\cot 2\theta$

$\frac{\sin 5\theta }{\cos 5\theta }=\frac{\cos 2\theta }{\sin 2\theta }$

$\sin 5\theta \sin 2\theta =\cos 5\theta \cos 2\theta$

$\cos 5\theta \cos 2\theta -\sin 5\theta \sin 2\theta =0$

$\cos \left(5\theta +2\theta \right)=0$

$\cos 7\theta =\cos \frac{\pi }{2}$

$7\theta =2n\pi \pm \frac{\pi }{2}$

$\theta =\frac{2n\pi }{7}\pm \frac{\pi }{14}$