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how do you prove

Can someone help me verify this trig identity?
(sin(x))/(1-cos(x))-(sin(x)cos(x))/(1+cos(x))=csc(x)(1+cos^2(x))

O I appluad you on your use of grouping symbols. Few do that when they post a problem.

$\frac{sin(x)}{1-cos(x)}-\frac{sin(x)cos(x)}{1+cos(x)}$

Cross multiply and simplify:

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

$\frac{sin(x)(1+cos^{2}(x))}{sin^{2}(x)}$

$\frac{1+cos^{2}(x)}{sin(x)}$

$csc(x)(1+cos^{2}(x))$

Comment on galactus's post

how did you get (1+cos^2(x)) in the third step?

Factor out the sin(x) in the numerator.

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