Chapter 5, Problem 21RE

To determine

To simplify: the expression and check result numerically.

Answer to Problem 21RE

  $\tan \left(\frac{\pi }{2}-x\right)\sec x=\csc x$

Explanation of Solution

Given information:

Given expression

  $\tan \left(\frac{\pi }{2}-x\right)\sec x$ .

Calculation:

Apply one of the co function identities, $\tan \left(\frac{\pi }{2}-x\right)$ to write the given expression.

  $\tan \left(\frac{\pi }{2}-x\right)\sec x=\left(\cot x\right)\left(\sec x\right)$

Apply one of the quotient identity, $\cot x=\frac{\cos x}{\sin x}$ .

  $\left(\cot x\right)\left(\sec x\right)=\left(\frac{\cos x}{\sin x}\right)\left(\sec x\right)$

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

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

Therefore, the expression simplifies to $\csc x$ .

Now, check the result graphically as shown:

Table of values for $\tan \left(\frac{\pi }{2}-x\right)\sec x$ is shown:

  

Graph of $\tan \left(\frac{\pi }{2}-x\right)\sec x$ is constructed as

  

Table of values for $\csc x$ is shown:

  

Graph of $\csc x$ is constructed as shown:

  

Both graphs are same and the functional values are also same.