# How do you prove cscx = sec(pi/2 - x)?

Mar 29, 2018

Since this is an identity,

the relation is true for all values of x

#### Explanation:

Given:

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

$\csc x = \frac{1}{\sin} x$

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

Thus,

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

$\cos \left(\frac{\pi}{2} - x\right) = \sin x$

Since this is an identity,

the relation is true for all values of x

Mar 29, 2018

See below

#### Explanation:

Using:
$\sec x = \frac{1}{\cos} x$
$\frac{1}{\sin} x = \csc x$
$\cos \left(x - y\right) = \cos x \cos y + \sin x \sin y$

Start:
$\csc x = \sec \left(\frac{\pi}{2} - x\right)$

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

$\csc x = \frac{1}{\cos \left(\frac{\pi}{2}\right) \cos x + \sin \left(\frac{\pi}{2}\right) \cdot \sin x}$

$\csc x = \frac{1}{\cancel{0 \cdot \cos x} + 1 \cdot \sin x}$

$\csc x = \frac{1}{\sin} x$

$\csc x = \csc x$