Prove (sinx/1-cosx)=cscx+cotx ?

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Prove (sinx/1-cosx)=cscx+cotx ?

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Answer 1 · 2018-04-03T14:09:34

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Explanation:

Left Hand Side:
$= \frac{sin x}{1 - cos x} (\frac{1 + cos x}{1 + cos x})$ $=sin x/(1-cos x)((1+cos x)/(1+cos x))$ -multiply by the conjugate

$= \frac{sin x + sin x cos x}{1 - {cos}^{2} x}$ $=(sin x + sin x cos x)/(1-cos^2x)$ -distribute

$= \frac{sin x}{{sin}^{2} x} + \frac{sin x cos x}{{sin}^{2} x}$ $=sin x / sin^2 x + ( sin x cos x )/ sin ^2 x$ -use property ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x + cos^2 x =1$

$= \frac{1}{sin x} + \frac{cos x}{sin x}$ $=1/ sin x + cos x / sin x$ -simply

$= csc x + cot x$ $=csc x + cot x$

$=$ $=$ Right Hand Side

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Prove (sinx/1-cosx)=cscx+cotx ?

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