Question

Evaluate:
${\sin }^{-1}\left\{\frac{\sin x+\cos x}{\sqrt{2}}\right\},-\frac{\pi }{4}<x<\frac{\pi }{4}.$

Answer 1

$si{n}^{-1}\left\{\frac{sinxcosx}{\sqrt{2}}\right\}$

$si{n}^{-1}\{\frac{sinx}{\sqrt{2}}+\frac{cosx}{\sqrt{2}}\}$

$si{n}^{-1}\{cos\frac{\pi }{4}sinx+sin\frac{\pi }{4}cosx\}$ $[\because sin\frac{\pi }{4}=cos\frac{\pi }{4}=\frac{1}{\sqrt{2}}]$

$si{n}^{-1}\left\{sin\right(x+\frac{\pi }{4}\left)\right\}$ $[\because sin(a+b)=sinacosb+sinbcosa]$

$x+\frac{\pi }{4}$ $[\because si{n}^{-1}(sinx)=x]$