Chapter 5, Problem 19RE

To determine

To simplify: the expression and check result numerically.

Answer to Problem 19RE

  ${\tan }^2\theta \left({\csc }^2\theta -1\right)=1$

Explanation of Solution

Given information:

Given expression

  ${\tan }^2\theta \left({\csc }^2\theta -1\right)$ .

Calculation:

Let us consider the following expression, ${\tan }^2\theta \left({\csc }^2\theta -1\right)$

Take out the common factor, ${\tan }^2\theta$ from both the terms

  ${\tan }^2\theta {\csc }^2\theta -{\tan }^2\theta ={\tan }^2\theta \left({\csc }^2\theta -1\right)$

Use the Pythagorean identity

  ${\csc }^2\theta -1={\cot }^2\theta$

Therefore, the above expression reduces to

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

Use the Reciprocal identity, $\cot \theta =\frac{1}{\tan \theta }$ to get

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

Therefore, the value of the expression ${\tan }^2\theta \left({\csc }^2\theta -1\right)$ is 1.

Now, check the result graphically as shown:

Table of values for ${\tan }^2\theta \left({\csc }^2\theta -1\right)$ is shown:

  

Graph of ${\tan }^2\theta \left({\csc }^2\theta -1\right)$ is constructed as

  

The functional values are same.