Question

If ${\sin }^{-1}\left(\frac{1}{3}\right)+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2},$ then find x.

Answer 1

We know that ${\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}$.
We have
$$\begin{gathered} {\sin }^{-1}\left(\frac{1}{3}\right)+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \Rightarrow {\sin }^{-1}\left(\frac{1}{3}\right)=\frac{\mathrm{\pi }}{2}-{\cos }^{-1}x \\ \Rightarrow {\sin }^{-1}\left(\frac{1}{3}\right)={\sin }^{-1}x \\ \Rightarrow x=\frac{1}{3} \end{gathered}$$

∴ $x=\frac{1}{3}$