Question

Simplify ${\sin }^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right)=x+\frac{\pi }{m},(\frac{-\pi }{4}<x<\frac{\pi }{4})$.Find $m$

Answer 1

${\sin }^{-1}\left(\frac{\sin x+\cos x}{\sqrt{2}}\right)$

$\Rightarrow {\sin }^{-1}(\frac{1}{\sqrt{2}}\sin x+\frac{1}{\sqrt{2}}\cos x)$

$\Rightarrow {\sin }^{-1}(\cos \frac{\pi }{4}\sin x+\sin \frac{\pi }{4}\cos x)$

$\Rightarrow {\sin }^{-1}\left[\sin \right(\frac{\pi }{4}\left)\right]=\frac{\pi }{4}+x$

On comparing with $x+\frac{\pi }{m}$, we have

$m=4$