Simplify the denominator

Use the identity $\sin^{2}\theta = 1 - \cos^{2}\theta$ to rewrite the denominator $1 + \cos\theta - \sin^{2}\theta$.

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

Substitute back into the expression

Replace the original denominator with the simplified form to cancel common terms.

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

Final simplification

Cancel the common term $(1 + \cos\theta)$ from the numerator and denominator.

$\frac{\sin\theta}{\cos\theta} = \tan\theta$