Chapter 13.6, Problem 48WE

To determine

To simplify: The each expression.

Answer to Problem 48WE

The simplified form of expression is $2{\sec }^{2}x$ .

Explanation of Solution

Given:

The given expression is as:

  ${\left(1+\tan x\right)}^2+{\left(1-\tan x\right)}^2$ .

The simplified form is calculated as:

  $\begin{array}{c}{\left(1+\tan x\right)}^2+{\left(1-\tan x\right)}^2=1+{\tan }^{2}x+2\tan x+1+{\tan }^{2}x-2\tan x\\ =2\left(1+{\tan }^{2}x\right)\\ =2{\sec }^{2}x\left[\because 1+{\tan }^{2}x={\sec }^{2}x\right]\end{array}$

Therefore the simplified form of expression is $2{\sec }^{2}x$ .