Chapter 14, Problem 9CLR

To determine

To write: The $\frac{\cos \theta }{{\sin }^3\theta }$ in terms of $\cot \theta$ .

Answer to Problem 9CLR

The $\frac{\cos \theta }{{\sin }^3\theta }$ in terms of $\cot \theta$ is $\cot \theta +{\cot }^3\theta$ .

Explanation of Solution

Given:

The expression is $\frac{\cos \theta }{{\sin }^3\theta }$ .

Calculation:

Simplify the expression as follows.

  $\begin{array}{c}\frac{\cos \theta }{{\sin }^3\theta }=\frac{\cos \theta }{\sin \theta }\cdot \frac{1}{{\sin }^2\theta }\\ =\cot \theta \cdot {\csc }^2\theta \\ =\cot \theta \left(1+{\cot }^2\theta \right)\\ =\cot \theta +{\cot }^3\theta \end{array}$

Therefore, the $\frac{\cos \theta }{{\sin }^3\theta }$ in terms of $\cot \theta$ is $\cot \theta +{\cot }^3\theta$ .