Question

If $\sin \left(\frac{\pi }{4}\cot \theta \right)=\cos \left(\frac{\pi }{4}\tan \theta \right)$, then find the general solution of $\theta$.

Answer 1

Given, $\sin \left(\frac{\pi }{4}\cot \theta \right)=\cos \left(\frac{\pi }{4}\tan \theta \right)$

$\Rightarrow \sin \left(\frac{\pi }{4}\cot \theta \right)=\sin (\frac{\pi }{2}-\frac{\pi }{4}\tan \theta )$

$\Rightarrow \frac{\pi }{4}\cot \theta =\frac{\pi }{2}-\frac{\pi }{4}\tan \theta$

$\Rightarrow \frac{\pi }{4}(\cot \theta +\tan \theta )=\frac{\pi }{2}$

$\Rightarrow \tan \theta +\frac{1}{\tan \theta }=2$

$\Rightarrow (\tan \theta -1{)}^2=0$

We know $\tan \theta =1=\tan \frac{\pi }{4}$

General solution for $\theta =n\pi +\frac{\pi }{4}$.