11) simplify the expression as much as possible. \n\frac{\\sin^{2}x}{1 - \\cos x}
11) simplify the expression as much as possible. \n\frac{\\sin^{2}x}{1 - \\cos x}
Answer
Explanation:
Step1: Use trig identity
Recall $\sin^{2}x=1 - \cos^{2}x$. So the expression becomes $\frac{1-\cos^{2}x}{1 - \cos x}$.
Step2: Factor the numerator
Factor $1-\cos^{2}x$ using $a^{2}-b^{2}=(a + b)(a - b)$. Here $a = 1$ and $b=\cos x$, so $1-\cos^{2}x=(1+\cos x)(1 - \cos x)$. The expression is now $\frac{(1+\cos x)(1 - \cos x)}{1 - \cos x}$.
Step3: Cancel out common factor
Cancel out the common factor $(1 - \cos x)$ (assuming $\cos x\neq1$). We get $1+\cos x$.
Answer:
$1+\cos x$