Question

Find the value of ${\sec }^{-1}x+cose{c}^{-1}x$.

Answer 1

Let $se{c}^{-1}x=y$ $y\in [0,\pi ]$

$\Rightarrow x=secy$

$=cosec(\frac{\pi }{2}-y)$

$y\in [0,\pi ]\Rightarrow \frac{\pi }{2}-y\in [\frac{-\pi }{2},\frac{\pi }{2}]$

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

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

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