Chapter 5.1, Problem 76E

Solution

To find: Whether the statement is true or false.

The statement is false.

Given information:

It is given that the identity $\sin \theta =\tan \theta \cos \theta$ is the set of all real numbers.

Formula used:

As per the Quotient Identities, it is known that $\tan \theta =\frac{\sin \theta }{\cos \theta }$ .

Calculation:

Divide both sides of the given equation with $\cos \theta$ .

  $\begin{array}{c}\frac{\sin \theta }{\cos \theta }=\frac{\tan \theta \cos \theta }{\cos \theta }\\ \tan \theta =\tan \theta \end{array}$

Since both sides is divided with $\cos \theta$ then,

  $\begin{array}{c}\cos \theta \ne 0\\ \theta \ne \frac{\pi }{2}+k\cdot \pi \end{array}$

Here, $k$ is integer and represents number of periods.

This statement is false because domain of validity for the identity are all real numbers except $\theta =\frac{\pi }{2}+k\cdot \pi$ .

Therefore, the statement is false.