Chapter 5, Problem 17CR

To determine

To determine: The simplified form of the expression using the fundamental identities.

Answer to Problem 17CR

The required expression is $\sin \theta$ .

Explanation of Solution

Given information:

The expression: $\frac{\tan \left(-\theta \right)}{\sec \theta }$

Formula used:

Trigonometric formulas:

  $\tan \left(-\theta \right)=\tan \theta$

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

  $\frac{1}{\sec \theta }=\cos \theta$

Calculation:

The given expression is $\frac{\tan \left(-\theta \right)}{\sec \theta }$ .

Simplify the above expression using the formula $\tan \left(-\theta \right)=\tan \theta$ , $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\frac{1}{\sec \theta }=\cos \theta$ .

  $\begin{array}{c}\frac{\tan \left(-\theta \right)}{\sec \theta }=\frac{\tan \theta }{\sec \theta }\\ =\frac{\frac{\sin \theta }{\cos \theta }}{\frac{1}{\cos \theta }}\\ =\frac{\sin \theta }{\cancel{\cos \theta }}\times \cancel{\cos \theta }\\ =\sin \theta \end{array}$

Hence, the required expression is $\sin \theta$ .