Chapter 5.1, Problem 25E

Solution

To simplify: The given expression $\left({\sec }^{2}x+{\csc }^{2}x\right)-\left({\tan }^{2}x+{\cot }^{2}x\right)$ to either a constant or a basic trigonometric function.

The simplified form of expression $\left({\sec }^{2}x+{\csc }^{2}x\right)-\left({\tan }^{2}x+{\cot }^{2}x\right)$ is $2$ .

Given information: The given expression is $\left({\sec }^{2}x+{\csc }^{2}x\right)-\left({\tan }^{2}x+{\cot }^{2}x\right)$ .

Formula used: The Pythagorean identities of the trigonometric function are as follows:

  $\begin{array}{l}\left({\sec }^2\theta -{\tan }^2\theta \right)=1\\ \left({\csc }^2\theta -{\cot }^2\theta \right)=1\end{array}$

Calculation:

Simplify the given expression by using Pythagorean identities.

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

Draw the graph of $\left({\sec }^{2}x+{\csc }^{2}x\right)-\left({\tan }^{2}x+{\cot }^{2}x\right)$ and 2.

  

Figure $\left(1\right)$

It can be observed that the graph of both function is same.

Thus, the simplified form of the expression $\left({\sec }^{2}x+{\csc }^{2}x\right)-\left({\tan }^{2}x+{\cot }^{2}x\right)$ is $2$ .