Question

If ${\cos }^{-1}x+{\cos }^{-1}y=\frac{\mathrm{\pi }}{4}$, find the value of ${\sin }^{-1}x+{\sin }^{-1}y$

Answer 1

$$\begin{gathered} {\cos }^{-1}x+{\cos }^{-1}y=\frac{\mathrm{\pi }}{4} \\ \Rightarrow \frac{\mathrm{\pi }}{2}-{\sin }^{-1}x+\frac{\mathrm{\pi }}{2}-{\sin }^{-1}y=\frac{\mathrm{\pi }}{4}\left[\because {\cos }^{-1}x=\frac{\mathrm{\pi }}{2}-{\sin }^{-1}x\right] \\ \Rightarrow \mathrm{\pi }-\left({\sin }^{-1}x+{\sin }^{-1}y\right)=\frac{\mathrm{\pi }}{4} \\ \Rightarrow {\sin }^{-1}x+{\sin }^{-1}y=\frac{3\mathrm{\pi }}{4} \end{gathered}$$