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

Sep 10, 2016

Proof below

Explanation:

First we will prove $1 + {\tan}^{2} \theta = {\sec}^{2} \theta$:
${\sin}^{2} \theta + {\cos}^{2} \theta = 1$
${\sin}^{2} \frac{\theta}{\cos} ^ 2 \theta + {\cos}^{2} \frac{\theta}{\cos} ^ 2 \theta = \frac{1}{\cos} ^ 2 \theta$
${\tan}^{2} \theta + 1 = {\left(\frac{1}{\cos} \theta\right)}^{2}$
$1 + {\tan}^{2} \theta = {\sec}^{2} \theta$

Now we can prove your question:
${\sec}^{4} \theta$
$= {\left({\sec}^{2} \theta\right)}^{2}$
$= {\left(1 + {\tan}^{2} \theta\right)}^{2}$
$= 1 + 2 {\tan}^{\theta} + {\tan}^{4} \theta$