Question

Differentiate

${\sin }^{-1}\left\{\frac{\sin x+\cos x}{\sqrt{2}}\right\},-\frac{3\pi }{4}<x<\frac{\pi }{4}$

Answer 1

$$\begin{gathered} \mathrm{Let},y={\sin }^{-1}\left\{\frac{\sin x+\cos x}{\sqrt{2}}\right\} \\ \Rightarrow y={\sin }^{-1}\left\{\sin x\left(\frac{1}{\sqrt{2}}\right)+\cos x\left(\frac{1}{\sqrt{2}}\right)\right\} \\ \Rightarrow y={\sin }^{-1}\left\{\sin x\cos \frac{\mathrm{\pi }}{4}+\cos x\sin \frac{\mathrm{\pi }}{4}\right\} \\ \Rightarrow y={\sin }^{-1}\left\{\sin \left(x+\frac{\mathrm{\pi }}{4}\right)\right\}...\left(\mathrm{i}\right) \\ \mathrm{Here},\frac{-3\mathrm{\pi }}{4}<x<\frac{\mathrm{\pi }}{4} \\ \Rightarrow \frac{-3\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{4}<x+\frac{\mathrm{\pi }}{4}<\frac{\mathrm{\pi }}{4}+\frac{\mathrm{\pi }}{4} \\ \Rightarrow \frac{-\mathrm{\pi }}{2}<x+\frac{\mathrm{\pi }}{4}<\frac{\mathrm{\pi }}{2} \\ \mathrm{From}\left(\mathrm{i}\right)\mathrm{we}\mathrm{get}, \\ \Rightarrow y=x+\frac{\mathrm{\pi }}{4}\left[\mathrm{Since},{\sin }^{-1}\left(\sin \theta \right)=\theta ,\mathrm{if}\theta \in \left[\frac{-\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right]\right] \\ \mathrm{Differentiating}\mathrm{it}\mathrm{with}\mathrm{respect}\mathrm{to}x, \\ \frac{dy}{dx}=1+0 \\ \therefore \frac{dy}{dx}=1 \end{gathered}$$