# How do you verify cscx-sinx=cosxcotx?

Apr 16, 2016

Let's start by stating the identities that will be important to this problem:

#### Explanation:

The reciprocal identity (1): $\csc x = \frac{1}{\sin} x$

The quotient identity (1): $\cot x = \cos \frac{x}{\sin} x$

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

Placing the left side on a common denominator:

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

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

Applying the Pythagorean identity ${\cos}^{2} x + {\sin}^{2} x = 1$, we get:

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

Hopefully this helps!