Answer
$2~cos^2~\theta-1 = \frac{1-tan^2~\theta}{1+tan^2~\theta}$
Work Step by Step
$2~cos^2~\theta-1 = 2~cos^2~\theta-(sin^2~\theta+cos^2~\theta)$
$2~cos^2~\theta-1 = cos^2~\theta-sin^2~\theta$
$2~cos^2~\theta-1 = (cos~\theta-sin~\theta)(cos~\theta+sin~\theta)$
$2~cos^2~\theta-1 = \frac{\frac{(cos~\theta-sin~\theta)}{cos~\theta}~~\frac{(cos~\theta+sin~\theta)}{cos~\theta}}{\frac{1}{cos^2~\theta}}$
$2~cos^2~\theta-1 = \frac{(1-tan~\theta)~(1+tan~\theta)}{\frac{cos^2~\theta+sin^2~\theta}{cos^2~\theta}}$
$2~cos^2~\theta-1 = \frac{1-tan^2~\theta}{1+tan^2~\theta}$
