# How do you verify the identity 1-tan^4theta=2sec^2theta-sec^4theta?

Sep 7, 2016

See the Explanation.

#### Explanation:

We will use the Identity $: {\sec}^{2} x = 1 + {\tan}^{2} x$.

$\text{The R.H.S.} = 2 {\sec}^{2} \theta - {\sec}^{4} \theta$

$= {\sec}^{2} \theta \left(2 - {\sec}^{2} \theta\right)$

$= \left(1 + {\tan}^{2} \theta\right) \left\{1 + \left(1 - {\sec}^{2} \theta\right)\right\}$

$= \left(1 + {\tan}^{2} \theta\right) \left\{1 - \left({\sec}^{2} \theta - 1\right)\right\}$

$= \left(1 + {\tan}^{2} \theta\right) \left(1 - {\tan}^{2} \theta\right)$

$= {1}^{2} - {\left({\tan}^{2} \theta\right)}^{2}$

$= 1 - {\tan}^{4} \theta$

$\text{=The L.H.S.}$