Chapter 5, Problem 20RE

To determine

To simplify: the expression and check result numerically.

Answer to Problem 20RE

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

Explanation of Solution

Given information:

Given expression

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

Calculation:

Use one of the Pythagorean identities, ${\sin }^{2}x+{\cos }^{2}x=1$ . Replace $1-{\cos }^{2}x={\sin }^{2}x$ in the expression.

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

By the reciprocal identity, $\sin x=\frac{1}{\csc x}$ .

  $\begin{array}{c}{\csc }^{2}x{\sin }^{2}x={\csc }^2\left(\frac{1}{{\csc }^{2}x}\right)\\ =1\end{array}$

Therefore, the expression simplifies to 1.

Now, check the result graphically as shown:

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

  

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

  

The functional values of both expression are same.