Chapter 7.4, Problem 24AYU

establish each identity.

$\sin \theta \left(\cot \theta +\tan \theta \right)=\sec \theta$

To determine

To find: The trigonometric expression $\text{sin}\theta (\text{cot}\theta +\text{tan}\theta )=\text{sec}\theta$.

Answer to Problem 24AYU

Solution:

The trigonometric expression $\text{sin}\theta (\text{cot}\theta +\text{tan}\theta )=\text{sec}\theta$ is established.

Explanation of Solution

Given:

The trigonometric expression $\text{sin}\theta (\text{cot}\theta +\text{tan}\theta )=\text{sec}\theta$.

Calculation:

Left hand side.

$=\text{sin}\theta (\text{cot}\theta +\text{tan}\theta )$

Quotient identity of $\text{tan}\theta$ and $\text{cot}\theta$ function.

$=\text{sin}\theta \left(\frac{\text{cos}\theta }{\text{sin}\theta }+\frac{\text{sin}\theta }{\text{cos}\theta }\right)$

Adding up the quotients.

$=\text{sin}\theta \left(\frac{{\text{cos}}^2\theta +{\text{sin}}^2\theta }{(\text{sin}\theta ).(\text{cos}\theta )}\right)$

By Pythagorean identity.

$=\frac{\text{sin}\theta }{(\text{sin}\theta ).(\text{cos}\theta )}$

Cancelling the common factor $\text{cos}\theta$.

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

By reciprocal identity.

$= \text{sec}\theta$ Right hand side.