Chapter 7.5, Problem 34E

To determine

To find: x by solving the equation for $0\le x<2\pi$ .

Answer to Problem 34E

  $\frac{\pi }{6},\frac{5\pi }{6}\text{ and }\frac{3\pi }{2}$

Explanation of Solution

Given information:

  ${\cot }^{2}x-\csc x=1$

Calculation:

The value of x for the given equation can be evaluated as follows:

  $\begin{array}{c}{\cot }^{2}x-\csc x=1\end{array}$

  $\begin{array}{c}{\csc }^{2}x-1-\csc x=1\left[\text{Using Pythagorean identity}\right]\end{array}$

  $\begin{array}{c}{\csc }^{2}x-\csc x-2=0\left[\text{Subtract 1 from each side}\right]\end{array}$

  $\begin{array}{c}\left(\csc x+1\right)\left(\csc x-2\right)=0\left[\text{Factors}\right]\end{array}$

  $\begin{array}{c}\csc x=2\text{ or }-1\end{array}$

  $\begin{array}{c}\text{First case:}\end{array}$

  $\begin{array}{c}\text{}\csc x=-1\end{array}$

  $\begin{array}{c}\frac{1}{\sin x}=-1\left[\text{Using reciprocal identity}\right]\end{array}$

  $\begin{array}{c}\sin x=-1\left[\text{Multiply each side by }-\sin x\right]\end{array}$

  $\begin{array}{c}x=\frac{3\pi }{2}\end{array}$

  $\begin{array}{c}\text{Second case:}\end{array}$

  $\begin{array}{c}\text{}\csc x=2\end{array}$

  $\begin{array}{c}\frac{1}{\sin x}=2\left[\text{Using reciprocal identity}\right]\end{array}$

  $\begin{array}{c}\sin x=\frac{1}{2}\left[\text{Multiply each side by }\frac{\sin x}{2}\right]\end{array}$

  $\begin{array}{c}x=\frac{\pi }{6},\frac{5\pi }{6}\end{array}$

  $\begin{array}{c}\end{array}$

Thus the values of x are $\frac{\pi }{6},\frac{5\pi }{6}\text{ and }\frac{3\pi }{2}$ .