# How do you simplify (1- tan^2θ) /( 1+ tan^2 θ)?

Mar 25, 2018

$2 {\cos}^{2} \theta - 1$ or $\cos 2 \theta$

#### Explanation:

Using the identities:
$1 + {\tan}^{2} \theta = {\sec}^{2} \theta$
$\frac{1}{\sec} \theta = \cos \theta$
$\tan \theta = \sin \frac{\theta}{\cos} \theta$
${\sin}^{2} \theta = 1 - {\cos}^{2} \theta$
$2 {\cos}^{2} \theta - 1 = \cos 2 \theta$

Start:
$\frac{1 - {\tan}^{2} \theta}{1 + {\tan}^{2} \theta} =$

$\frac{1 - {\tan}^{2} \theta}{{\sec}^{2} \theta} =$

Split the numerator:

$\frac{1}{\sec} ^ 2 \theta - {\tan}^{2} \frac{\theta}{\sec} ^ 2 \theta =$

${\cos}^{2} \theta - {\sin}^{2} \frac{\theta}{\cancel{{\cos}^{2} \theta}} \cdot \cancel{{\cos}^{2} \theta} =$

${\cos}^{2} \theta - \left(1 - {\cos}^{2} \theta\right) =$

$2 {\cos}^{2} \theta - 1 =$

$\cos 2 \theta$

Mar 25, 2018

$\cos 2 \theta$

#### Explanation:

$\text{using the "color(blue)"trigonometric identities}$

•color(white)(x)tantheta=sintheta/costheta

•color(white)(x)sin^2theta+cos^2theta=1

•color(white)(x)cos^2theta-sin^2theta=cos2theta

$\Rightarrow \frac{1 - {\tan}^{2} \theta}{1 + {\tan}^{2} \theta}$

$= \frac{1 - {\sin}^{2} \frac{\theta}{\cos} ^ 2 \theta}{1 + {\sin}^{2} \frac{\theta}{\cos} ^ 2 \theta} \times {\cos}^{2} \frac{\theta}{\cos} ^ 2 \theta$

$= \frac{{\cos}^{2} \theta - {\sin}^{2} \theta}{{\cos}^{2} \theta + {\sin}^{2} \theta}$

$= {\cos}^{2} \theta - {\sin}^{2} \theta = \cos 2 \theta$