# How do you simplify (2-sec^2x)/(sec^2x)?

Apr 6, 2018

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

#### Explanation:

First, split the fraction.

$\frac{2 - {\sec}^{2} \left(x\right)}{\sec} ^ 2 \left(x\right)$

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

Because $\cos \left(x\right) = \frac{1}{\sec} \left(x\right)$, $\textcolor{red}{{\cos}^{2} \left(x\right) = \frac{1}{\sec} ^ 2 \left(x\right)}$

$= \frac{2}{\textcolor{red}{{\sec}^{2} \left(x\right)}} - {\sec}^{2} \frac{x}{\sec} ^ 2 \left(x\right)$

$= \textcolor{b l u e}{2 {\cos}^{2} \left(x\right) - 1}$

Observe the following:

$\textcolor{b l u e}{\cos} \left(2 x\right)$

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

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

$= \textcolor{b l u e}{2 {\cos}^{2} \left(x\right) - 1}$

Therefore,

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

Apr 6, 2018

It simplifies to $\cos \left(2 x\right)$.

#### Explanation:

Use these identites:

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

$\sec x = \frac{1}{\cos} x q \quad \textcolor{b l u e}{\implies} q \quad {\sec}^{2} x = \frac{1}{\cos} ^ 2 x$

First, split the fraction:

$\textcolor{w h i t e}{=} \frac{2 - {\sec}^{2} x}{\sec} ^ 2 x$

$= \frac{2}{\sec} ^ 2 x - {\sec}^{2} \frac{x}{\sec} ^ 2 x$

$= \frac{2}{\sec} ^ 2 x - 1$

$= 2 \cdot \frac{1}{\sec} ^ 2 x - 1$

$= 2 \cdot \frac{1}{\frac{1}{\cos} ^ 2 x} - 1$

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

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

$= \cos \left(2 x\right)$

Hope this helped!