Chapter 5.1, Problem 23E

Solution

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

The simplified form of the expression $\frac{\tan \left(\frac{\pi }{2}-x\right)\csc x}{{\csc }^{2}x}$ is $\cos x$ .

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

Formula used: The cofunction identity of trigonometric function is $\tan \left(\frac{\pi }{2}-\theta \right)=\cot \theta$ .

Calculation:

Simplify the given expression using tangent cofunction identity.

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

Apply reciprocal identity $\cot x=\frac{\cos x}{\sin x}$ .

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

Draw the graph of $\frac{\tan \left(\frac{\pi }{2}-x\right)\csc x}{{\csc }^{2}x}$ and $\cos x$ .

  

Figure (1)

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

Thus, the simplified form of the expression $\frac{\tan \left(\frac{\pi }{2}-x\right)\csc x}{{\csc }^{2}x}$ is $\cos x$ .