# How do I prove this equation is an identity? sin^3+sin(x)cos^2(x)/cos(x)=tan(x)

Mar 18, 2018

Verified below

#### Explanation:

$\frac{{\sin}^{3} x + \sin \left(x\right) {\cos}^{2} \left(x\right)}{\cos} \left(x\right) = \tan \left(x\right)$

GCF:
$\frac{\sin x \left({\sin}^{2} x + {\cos}^{2} x\right)}{\cos} \left(x\right) = \tan \left(x\right)$

Apply the pythagorean identity:
${\sin}^{2} x + {\cos}^{2} x = 1$

Therefore:
$\frac{\sin x \left(1\right)}{\cos} \left(x\right) = \tan \left(x\right)$
$\sin \frac{x}{\cos} x = \tan \left(x\right)$

Apply the quotient identity:
$\sin \frac{x}{\cos} x = \tan x$

Therefore:
$\tan x = \tan x$