# How do i verify the following trig identity? (tan^2x + 1)(cos^2x - 1) = -tan^2x

May 25, 2018

See below.

#### Explanation:

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

${\sec}^{2} x = \frac{1}{\cos} ^ 2 x$

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

To find cos^2x -1, we can use the identity ${\sin}^{2} x + {\cos}^{2} x = 1$.

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

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

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

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