Chapter 7.4, Problem 52AYU

establish each identity.

$1-\frac{{\sin }^2\theta }{1+\cos \theta }=\cos \theta$

To determine

To find: The trigonometric expression $1- \frac{{\text{sin}}^2\theta }{1+ \cos \theta }= \cos \theta$.

Answer to Problem 52AYU

Solution:

The trigonometric expression $1- \frac{{\text{sin}}^2\theta }{1+ \cos \theta }= \cos \theta$ is established.

Explanation of Solution

Given:

The trigonometric expression $1- \frac{{\text{sin}}^2\theta }{1+ \cos \theta }= \cos \theta$.

Calculation:

Left hand side,

$1- \frac{{\text{sin}}^2\theta }{1+ \cos \theta }$

By Pythagorean identity.

$1-\left(\frac{1- {\text{cos}}^2\theta }{1+\cos \theta }\right)$

Expanding,

$1- \frac{(1- \cos \theta )(1+\cos \theta )}{1+\cos \theta }$

Cancelling the common factor.

$1-(1-\cos \theta )$

$\cos \theta$

$\text{LHS }=\text{ RHS}$