# How do you prove cos2x= (1-tan^2(x))/ (1+tan^2(x))?

Apr 16, 2015

In this way:

(remembering that $\tan x = \sin \frac{x}{\cos} x$ and ${\sin}^{2} x + {\cos}^{2} x = 1$),

the second member becomes:

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

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

$= {\cos}^{2} x - {\sin}^{2} x$,

that is the development of the formula of $\cos 2 x$.