Prove the following identity: cot x= sin 2x/(1-cos2x)?

Work on LHS: sin 2x = 2 sin x cos c cot x = (cos x) / (sin x) sin 2x - cot x = (2 sin x cos x) - (cos x)/(sin x) = 2 sin^2 x cos x - cos x / (sin x) cos x * (2 sin^2 x - 1) / (sin x) 2 sin^2 x - 1 = - (1 - 2 sin^2 x) = - cos (2x) so... (cos x)/(sin x) * (- cos 2x) = - cot x cos (2x) qed.

Simplifying the left hand side: sin(2x) - cot(x) = 2sin(x)cos(x) - (cos(x))/sin(x) = cos(x)(2sin(x) - 1/sin(x)) = cos(x)/sin(x) * (2sin^2(x) - 1) = -cot(x)(1 - 2sin^2(x)) = -cot(x)(1 - 2*(1 - cos(2x))/2) = -cot(x)*(1 - 1 + cos(2x)) = -cot(x)*cos(2x).

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