Chapter 5.1, Problem 117E

To determine

To express: The given expression in terms of $\sin \theta$ and $\cos \theta$ .

Answer to Problem 117E

The expression ( $\frac{\sec \theta (1+\tan \theta )}{\sec \theta +\csc \theta }$ ) can be expressed as $\left(\frac{\sin \theta }{\cos \theta }\right)$ .

Explanation of Solution

Given information:

The given expression:

  $\frac{\sec \theta (1+\tan \theta )}{\sec \theta +\csc \theta }$

Calculation:

As per problem,

The given expression:

  $\frac{\sec \theta (1+\tan \theta )}{\sec \theta +\csc \theta }$

Write both numerator and denominator in terms of $\sin \theta$ and $\cos \theta$ .

  $\begin{array}{l}\frac{\sec \theta (1+\tan \theta )}{\sec \theta +\csc \theta }=\frac{\left(\frac{1}{\cos \theta }\right)(1+\tan \theta )}{\left(\frac{1}{\cos \theta }\right)+\left(\frac{1}{\sin \theta }\right)}\text{ [Reciprocal identities: sec}\theta \text{=}\frac{1}{\cos \theta }\text{ and csc}\theta \text{=}\frac{1}{\sin \theta }]\\ =\frac{\left(\frac{1}{\cos \theta }\right)\left(1+\frac{\sin \theta }{\cos \theta }\right)}{\left(\frac{1}{\cos \theta }\right)+\left(\frac{1}{\sin \theta }\right)}\text{ [Quotient identity: tan}\theta \text{=}\frac{\sin \theta }{\cos \theta }]\\ =\frac{\left(\frac{1}{\cos \theta }\right)\left(\frac{\cos \theta +\sin \theta }{\cos \theta }\right)}{\left(\frac{\sin \theta +\cos \theta }{\cos \theta \sin \theta }\right)}\text{ [Add fractions]}\\ =\frac{\left(\frac{\cos \theta +\sin \theta }{{\cos }^2\theta }\right)}{\left(\frac{\sin \theta +\cos \theta }{\cos \theta \sin \theta }\right)}\text{ [Multiply]}\\ =\left(\frac{\cos \theta +\sin \theta }{{\cos }^2\theta }\right)\cdot \left(\frac{\cos \theta \sin \theta }{\sin \theta +\cos \theta }\right)\text{ [Write as product of fractions]}\\ =\left(\frac{\cancel{\cos \theta +\sin \theta }}{\cos \theta \cdot \cancel{\cos \theta }}\right)\cdot \left(\frac{\cancel{\cos \theta }\cdot \sin \theta }{\cancel{\sin \theta +\cos \theta }}\right)\text{ [Divide out common factors]}\\ =\frac{\sin \theta }{\cos \theta }\text{ [Simplify]}\end{array}$

Hence, the expression ( $\frac{\sec \theta (1+\tan \theta )}{\sec \theta +\csc \theta }$ ) can be expressed as $\left(\frac{\sin \theta }{\cos \theta }\right)$ .