# How do you prove Sec(x) - cos(x) = sin(x) * tan(x)?

May 30, 2016

Recall that $\sec \theta = \frac{1}{\cos} \theta$ and that $\tan \theta = \sin \frac{\theta}{\cos} \theta$

$\frac{1}{\cos} x - \cos x = \sin x \times \sin \frac{x}{\cos} x$

Now put the left side on a common denominator:

$\frac{1}{\cos} x - \frac{\cos x \times \cos x}{\cos} x = \sin x \times \sin \frac{x}{\cos} x$

Simplify:

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

Now use the modified pythagorean identity ${\sin}^{2} x + {\cos}^{2} x = 1$: ${\sin}^{2} \theta = 1 - {\cos}^{2} \theta$.

$\frac{{\sin}^{2} x}{\cos} x = {\sin}^{2} \frac{x}{\cos} x \to$ Identity proved!!

Practice exercises:

1. Prove the following identities, using the quotient, pythagorean and reciprocal identities.

a) $\frac{1}{\tan} x + \tan x = \frac{1}{\sin x \cos x}$

b) ${\cos}^{2} x = \frac{\csc x \cos x}{\tan x + \cot x}$

Hopefully this helps, and good luck!