Chapter 5.5, Problem 99E

To determine

Toproof:Theidentity $\csc 2\theta =\frac{\csc \theta }{2\cos \theta }$ algebraically and verify the result using graphical utility.

Explanation of Solution

Given information:

The given identity is $\csc 2\theta =\frac{\csc \theta }{2\cos \theta }$ .

Formula used:

  $\sin 2\theta =2\sin \theta \cos \theta$

  $\csc \theta =\frac{1}{\sin \theta }$

Proof:

Let take the left hand side of the identity and expand it.

  $\begin{array}{c}\csc 2\theta =\frac{1}{\sin 2\theta }\left\{\text{Reciprocal}\text{identity},\csc \theta =\frac{1}{\sin \theta }\right\}\\ =\frac{1}{2\sin \theta \cos \theta }\left\{\sin 2\theta =2\sin \theta \cos \theta \right\}\\ =\frac{\csc \theta }{2\cos \theta }\end{array}$

From the above, it is clear that the left hand side of the equation is equal to the right hand side of the equation.

Hence, proved.

To verify the result, draw the graph for $\csc 2\theta$ and $\frac{\csc \theta }{2\cos \theta }$ using graphical utility.

  

From the graph, it is clear that the graphs coincide. Hence, it is proved.