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If $\cos x \neq-1$, then what is $\frac{\sin x}{1+\cos x}$ equal to?
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Verified Answer
The correct answer is:
$\tan \frac{x}{2}$
Consider $\frac{\sin x}{1+\cos x}=\frac{2 \sin x / 2 \cos x / 2}{1+2 \cos ^{2}(x / 2)-1}$
$\left(\because \sin 2 x=2 \sin x \cos x\right.$ and $\left.\cos 2 x=2 \cos ^{2} x-1\right)$
$=\frac{2 \sin x / 2 \cos x / 2}{2 \cos ^{2} x / 2}=\frac{\sin x / 2}{\cos x / 2}=\tan x / 2$
$\left(\because \sin 2 x=2 \sin x \cos x\right.$ and $\left.\cos 2 x=2 \cos ^{2} x-1\right)$
$=\frac{2 \sin x / 2 \cos x / 2}{2 \cos ^{2} x / 2}=\frac{\sin x / 2}{\cos x / 2}=\tan x / 2$
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