Chapter 13.3, Problem 47SR

To determine

To verify: The given identity.

Explanation of Solution

Given:

The identity is given as:

  $\frac{\sin \theta }{\tan \theta }+\frac{\cos \theta }{\cot \theta }=\cos \theta +\sin \theta$

According to the quotient identity:

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

Now, considering the left hand side of the identity:

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

From the above result it is clear that the left hand side of the identity is equal to the right hand side of the identity.

Therefore, the identity is verified.