Question

$\sin \left({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x\right)=1$

Answer 1

$$\begin{gathered} \sin \left({\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x\right)=1 \\ \Rightarrow {\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x={\sin }^{-1}1 \\ \Rightarrow {\sin }^{-1}\frac{1}{5}+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\sin }^{-1}\frac{1}{5}=\frac{\mathrm{\pi }}{2}-{\cos }^{-1}x \\ \Rightarrow {\sin }^{-1}\frac{1}{5}={\sin }^{-1}x\left[\because {\sin }^{-1}x=\frac{\mathrm{\pi }}{2}-{\cos }^{-1}x\right] \\ \Rightarrow x=\frac{1}{5} \end{gathered}$$