Chapter 5, Problem 26RE

To determine

To show: that the rate of change can also be written as $\cot x\sqrt{\sin x}$ .

Explanation of Solution

Given information:

The rate of change of the function $f\left(x\right)=\csc x-\cot x$ is given by ${\csc }^{2}x-\csc x\cot x$ .

Calculation:

Consider the function

  ${\csc }^{2}x-\csc x\cot x$

We have to determine the equivalent form of the given expression in terms of the sine and cosine of $x$ . This can be done by the use of identities given by equation (1) and (2).

  $\begin{array}{l}\csc x=\frac{1}{\sin x}\mathrm{..................}(1)\\ \cot x=\frac{\cos x}{\sin x}\mathrm{.............}(2)\end{array}$

The given expression is simplified and finally represented in terms of $\sin x$ and $\cos x$ as follows by use identities given by equation (1) and (2).

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