Question

Write the value of $\cos \left({\sin }^{-1}x+{\cos }^{-1}x\right),\left|x\right|\le 1$

Answer 1

We have
$$\begin{gathered} \left|x\right|\le 1 \\ \Rightarrow \pm x\le 1 \\ \Rightarrow x\le 1\mathrm{or}-x\le 1 \\ \Rightarrow x\le 1\mathrm{or}x\ge -1 \\ \Rightarrow x\in \left[-1,1\right] \end{gathered}$$
Now,
$$\begin{gathered} \cos \left({\sin }^{-1}x+{\cos }^{-1}x\right)=\cos \left(\frac{\mathrm{\pi }}{2}\right)\left[\because {\sin }^{-1}x+{\cos }^{-1}x=\frac{\mathrm{\pi }}{2}\right] \\ =0 \end{gathered}$$