Question

Find the value of $\cos ({\sec }^{-1}x+{\csc }^{-1}x)$, $\left|x\right|\ge 1$

Answer 1

${\sec }^{-1}x={\cos }^{-1}\left(\frac{1}{x}\right)=\frac{\pi }{2}-{\sin }^{-1}\left(\frac{1}{x}\right)=\frac{\pi }{2}-cose{c}^{-1}x$

$\Rightarrow {\sec }^{-1}x+cose{c}^{-1}x=\frac{\pi }{2}$

$\Rightarrow \cos \left(\frac{\pi }{2}\right)=0$.