Chapter 7.4, Problem 13AYU

In Problems 11-20, simplify each trigonometric expression by following the indicated direction.

Rewrite over a common denominator: $\frac{\sin \theta +\cos \theta }{\cos \theta }+\frac{\cos \theta -\sin \theta }{\sin \theta }$

To determine

To find: Rewrite the trigonometric expression $\frac{\text{sin}\theta + \text{cos}\theta }{\text{cos}\theta }+ \frac{\text{cos}\theta -\text{sin}\theta }{\text{sin}\theta }$ over a common denominator.

Answer to Problem 13AYU

Solution:

The trigonometric expression $\frac{\text{sin}\theta + \text{cos}\theta }{\text{cos}\theta }+ \frac{\text{cos}\theta -\text{sin}\theta }{\text{sin}\theta }$ over a common denominator.

$=\sec \theta \csc \theta$

Explanation of Solution

Given:

The trigonometric expression $\frac{\text{sin}\theta + \text{cos}\theta }{\text{cos}\theta }+ \frac{\text{cos}\theta -\text{sin}\theta }{\text{sin}\theta }$.

Calculation:

Add the quotients.

$\frac{\text{sin}\theta (\text{sin}\theta + \text{cos}\theta )}{\text{cos}\theta }+\frac{\text{cos}\theta (\text{cos}\theta -\text{sin}\theta )}{\text{sin}\theta }$

Remove the parenthesis in the numerator.

$\frac{{\text{sin}}^2\theta +\text{sin}\theta .\text{cos}\theta }{\text{cos}\theta }+\frac{{\text{cos}}^2\theta -\text{cos}\theta .\text{sin}\theta }{\text{sin}\theta }$

Regroup.

$\frac{{\text{sin}}^2\theta +{\text{cos}}^2\theta +\text{sin}\theta .\text{cos}\theta -\text{cos}\theta .\text{sin}\theta }{\text{cos}\theta .\text{sin}\theta }$

Pythagorean identity.

$\frac{1}{\text{cos}\theta . \text{sin}\theta }$

$\frac{1}{\text{cos}\theta } . \frac{1}{\text{sin}\theta }$

$\text{sec}\theta \text{csc}\theta$