Question

If $\cos \left({\sin }^{-1}\frac{2}{5}+{\cos }^{-1}x\right)=0$, find the value of x.

Answer 1

$$\begin{gathered} \cos \left({\sin }^{-1}\frac{2}{5}+{\cos }^{-1}x\right)=0 \\ \Rightarrow \cos \left({\sin }^{-1}\frac{2}{5}+{\cos }^{-1}x\right)=\cos \left(\frac{\mathrm{\pi }}{2}\right) \\ \Rightarrow {\sin }^{-1}\frac{2}{5}+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2} \\ \therefore x=\frac{2}{5}\left[\because {\sin }^{-1}y+{\cos }^{-1}y=\frac{\mathrm{\pi }}{2}\right] \end{gathered}$$