Full solution

1

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

Distribute and Simplify: Now, simplify the expression by distributing the numerator over the denominator. $\newline$ $1 - \frac{1 - \cos^2(\theta)}{1 + \cos(\theta)} = 1 - \left(\frac{1}{1 + \cos(\theta)} - \frac{\cos^2(\theta)}{1 + \cos(\theta)}\right)$
Combine Terms: Combine the terms over a common denominator. $\newline$ $1 - \left(\frac{1}{1 + \cos(\theta)} - \frac{\cos^2(\theta)}{1 + \cos(\theta)}\right) = 1 - \frac{1 - \cos^2(\theta)}{1 + \cos(\theta)}$
Use Pythagorean Identity: Simplify the numerator by using the Pythagorean identity again, $1 - \cos^2(\theta) = \sin^2(\theta)$ . $\newline$ $1 - \frac{1 - \cos^2(\theta)}{1 + \cos(\theta)} = 1 - \frac{\sin^2(\theta)}{1 + \cos(\theta)}$
Subtract from $1$ : Now, subtract the fraction from $1$ . $\newline$ $1 - \frac{\sin^2(\theta)}{1 + \cos(\theta)} = \frac{1 + \cos(\theta)}{1 + \cos(\theta)} - \frac{\sin^2(\theta)}{1 + \cos(\theta)}$
Combine Terms: Combine the terms over the common denominator. $\newline$ $(1 + \cos(\theta))/(1 + \cos(\theta)) - (\sin^2(\theta))/(1 + \cos(\theta)) = (1 + \cos(\theta) - \sin^2(\theta))/(1 + \cos(\theta))$
Replace with Identity: Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$ to replace $\sin^2(\theta)$ with $1 - \cos^2(\theta)$ . $\frac{1 + \cos(\theta) - \sin^2(\theta)}{1 + \cos(\theta)} = \frac{1 + \cos(\theta) - (1 - \cos^2(\theta))}{1 + \cos(\theta)}$
Simplify Numerator: Simplify the numerator. $\newline$ $(1 + \cos(\theta) - (1 - \cos^2(\theta)))/(1 + \cos(\theta)) = (\cos(\theta) + \cos^2(\theta))/(1 + \cos(\theta))$
Factor Out: Factor out $\cos(\theta)$ from the numerator. $\frac{\cos(\theta) + \cos^2(\theta)}{1 + \cos(\theta)} = \frac{\cos(\theta)(1 + \cos(\theta))}{1 + \cos(\theta)}$
Cancel Common Terms: Cancel out the common terms in the numerator and the denominator. $\newline$ $\frac{\cos(\theta)(1 + \cos(\theta))}{1 + \cos(\theta)} = \cos(\theta)$