First you must write the expression as
1/(1 – Cos(x)) – 1/(1 + Cos(x)) = 2Csc(x)Cot(x)
You may have read 1 – cosX as a denominator, but nobody else will.
The common denominator of the left side is (1 – Cos(x))(1 + Cos(x))
So the left side is
[(1 + Cos(x) – (1 – Cos(x))]/[(1 – Cos(x))(1 + Cos(x))]
Which simplifies to
[2Cos(x)]/[1 - Cos²(x))] = [2Cos(x)]/[Sin²(x)]
= 2[Cos(x)/Sin(x)][1/Sin(x)]
= 2Cot(x)Csc(x) as required
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