# Question #788a3

May 24, 2017

By derivation.
Start with ${\cos}^{2} \left(x\right) + {\sin}^{2} \left(x\right) = 1$
Divide both sides by ${\cos}^{2} \left(x\right)$
The result is: $1 + {\sin}^{2} \frac{x}{\cos} ^ 2 \left(x\right) = \frac{1}{\cos} ^ 2 \left(x\right)$
This the same as the given equation.

May 24, 2017

Use trig properties.

#### Explanation:

We know that:

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

We also know that:

$\sec \theta = \frac{1}{\cos} \theta$

Transforming the second property listed, we get:

$\sec \theta = \frac{1}{\cos} \theta$
${\left(\sec \theta\right)}^{2} = {\left(\frac{1}{\cos} \theta\right)}^{2}$

Then applying the first property:

$1 + {\tan}^{2} \theta = {\sec}^{2} \theta = \frac{1}{\cos} ^ 2 \theta$

$Q E D$