# How do you verify (1 + sec^2 x) / (1 + tan^2 x) = 1 + cos^2 x?

Jul 3, 2016

Use the Pythagorean Identity $1 + {\tan}^{2} x = {\sec}^{2} x$.

#### Explanation:

Recall the Pythagorean Identity $1 + {\tan}^{2} x = {\sec}^{2} x$ (this can be derived by dividing the identity ${\sin}^{2} x + {\cos}^{2} x = 1$ by ${\cos}^{2} x$). The key to this problem is applying this identity.

Since $1 + {\tan}^{2} x = {\sec}^{2} x$, we can replace the $1 + {\tan}^{2} x$ in the denominator with ${\sec}^{2} x$:
$\frac{1 + {\sec}^{2} x}{{\sec}^{2} x} = 1 + {\cos}^{2} x$

Now we can break the fraction up in two:
$\frac{1}{\sec} ^ 2 x + {\sec}^{2} \frac{x}{\sec} ^ 2 x = 1 + {\cos}^{2} x$

Since $\frac{1}{\sec} x = \cos x$, $\frac{1}{\sec} ^ 2 x = {\cos}^{2} x$; and ${\sec}^{2} \frac{x}{\sec} ^ 2 x = 1$. So:
${\cos}^{2} x + 1 = 1 + {\cos}^{2} x$

Using the commutative property of addition we can rearrange the left side of the equation to match the right:
$1 + {\cos}^{2} x = 1 + {\cos}^{2} x$