# How do you prove sec^-1x+csc^-1x=pi/2?

Oct 20, 2016

#### Explanation:

Prove:

${\sec}^{-} 1 \left(x\right) + {\csc}^{-} 1 \left(x\right) = \frac{\pi}{2}$

Use the identity ${\csc}^{-} 1 \left(x\right) = \frac{\pi}{2} - {\sec}^{-} 1 \left(x\right)$:

${\sec}^{-} 1 \left(x\right) + \frac{\pi}{2} - {\sec}^{-} 1 \left(x\right) = \frac{\pi}{2}$

$\frac{\pi}{2} = \frac{\pi}{2}$

Q.E.D.

Oct 20, 2016

use the fact that csc is the complementary function of sec.

#### Explanation:

let
${\sec}^{-} 1 x = y$

$\implies x = \sec y$

$\implies x = \csc \left(\frac{\pi}{2} - y\right)$
$\implies {\csc}^{-} 1 x = \frac{\pi}{2} - y$

substituting back for y

${\csc}^{-} 1 x = \frac{\pi}{2} - {\sec}^{-} 1 x$

hence ${\sec}^{-} 1 x + {\csc}^{-} 1 x = \frac{\pi}{2}$

as required.