# How do you verify the identity cosxcotx = cscx - sinx?

Mar 17, 2018

All the identities you will need:
$\cot x = \cos \frac{x}{\sin} x$
${\cos}^{2} x = 1 - {\sin}^{2} x$
$\frac{1}{\sin} x = \csc x$

Starting:
$\cos x \cot x = \csc x - \sin x$

Apply number 1 on the list:

$\cos x \cdot \cos \frac{x}{\sin} x = \csc x - \sin x$

Simplify:

${\cos}^{2} \frac{x}{\sin} x = \csc x - \sin x$

Apply number 2 on the list:

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

Split the numerator:

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

Apply number 3 on the list:

$\csc x - \sin x = \csc x - \sin x$

Mar 17, 2018

Kindly go through a Proof in the Explanation.

#### Explanation:

We have, $\cos x \cot x + \sin x$,

$= \cos x \cdot \cos \frac{x}{\sin} x + \sin x$,

$= \frac{{\cos}^{2} x + {\sin}^{2} x}{\sin} x$,

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

$\therefore \cos x \cot x + \sin x = \csc x$.

$\Rightarrow \cos x \cot x = \csc x - \sin x$, as desired!