Chapter 5.2, Problem 50E

To determine

To verify the identity:

  $\frac{3\tan \theta }{9-9{\sec }^2\theta }=\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}$

And also verify this using table feature of graphing utility.

Explanation of Solution

Given:

  $\frac{3\tan \theta }{9-9{\sec }^2\theta }=\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}$

Concept Used:

The trigonometric identities:

 • $\sec x=\frac{1}{\cos x}$

 • ${\sin }^{2}x+{\cos }^{2}x=1$

Calculation:

In order to verify the identity:

  $\frac{3\tan \theta }{9-9{\sec }^2\theta }=\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}$

Take left hand of above identity and simplify it using the trigonometric identities.

As,

  $\begin{array}{c}\frac{3\tan \theta }{9-9{\sec }^2\theta }=\frac{3\tan \theta }{9\left(1-{\sec }^2\theta \right)}\\ =\frac{3\frac{\sin \theta }{\cos \theta }}{9\left(1-\frac{1}{{\cos }^2\theta }\right)}\\ =\frac{3\sin \theta \cos \theta }{9\left({\cos }^2\theta -1\right)}\\ =\frac{\sin \theta \cos \theta }{3\left({\cos }^2\theta -1\right)}\\ =\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}\\ =\text{Right hand}\end{array}$

Thus, identity is verified, i.e.

  $\frac{3\tan \theta }{9-9{\sec }^2\theta }=\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}$

Table of values for $\frac{3\tan \theta }{9-9{\sec }^2\theta }$ is shown:

  

Graph of $\frac{3\tan \theta }{9-9{\sec }^2\theta }$ is constructed as

  

Table of values for $\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}$ is shown:

  

Graph of $\frac{\cos \theta \sin \theta +\cos \theta }{3\left(\sin \theta +{\sin }^2\theta \right)}$ is constructed as shown:

  

Since, the graph of left hand side and right hand side of the identity coincides, thus the identity is verified by graph too.