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

We need to simplify $\frac{1-tan^{2}x}{1+tan^{2}x}$

## Solution

$\frac{1-tan^{2}x}{1+tan^{2}x}$

We know from trigonometric identity that

$1 + \tan ^{2}x= \sec^{2}x$

Hence the given equation becomes

$\frac{1-tan^{2}x}{sec^{2}x}$ $\frac{1}{sec^{2}x} – \frac{tan^{2}x}{sec^{2}x}$

We know that 1 / sec x = cos x and tan x can be expressed as sin x / cos x

=$\frac{cos^{2}}{x} – \frac{sin^{2}x}{cos^{2}x}$ . cos2x

= cos2x – sin2x

$\frac{1-tan^{2}x}{1+tan^{2}x}$ = cos2x – sin2x