Chapter 7.4, Problem 21AYU

establish each identity.

$1+{\tan }^2\left(-\theta \right)={\sec }^2\theta$

To determine

To find: The trigonometric expression $1+{\text{tan}}^2\left(-\theta \right)={\text{sec}}^2\theta$.

Answer to Problem 21AYU

Solution:

The trigonometric expression $1+{\text{tan}}^2\left(-\theta \right)={\text{sec}}^2\theta$ is established.

Explanation of Solution

Given:

The trigonometric expression $1+{\text{tan}}^2\left(-\theta \right)={\text{sec}}^2\theta$.

Calculation:

Left hand side.

$=1+{\text{tan}}^2\left(-\theta \right)$

Odd property of $\text{tan}$.

$= 1+(-\text{tan}\theta {)}^2$

$= 1+{\text{tan}}^2\theta$

By Pythagorean identity.

$={\text{sec}}^2\theta$ Right hand side.