# How do you verify the identity 2sec^2x-2sec^2xsin^2x-sin^2x-cos^2x=1?

Jan 19, 2017

Let's do a little bit of factoring.

$2 {\sec}^{2} x \left(1 - {\sin}^{2} x\right) - {\sin}^{2} x - {\cos}^{2} x = 1$

Use the identity ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$:

$2 {\sec}^{2} x \left({\cos}^{2} x\right) - {\sin}^{2} x - {\cos}^{2} x = 1$

Secant and cosine are inverses; their product is $1$.

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

You will want to convert all to sine or all to cosine, using the pythagorean identity given above.

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

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

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

$1 = 1$

$L H S = R H S$

The identity is proved.

Hopefully this helps!