Prove Cos2x 2 cot2 cos x21 cot xSolutionUsing the trignom

Prove: Cos(2x) = 2 cot^2 - cos (x^2)/1 + cot x

Solution

Using the trignometric double angle identity, we know that

Cos(2X) = Cos²X - Sin²X ......... (1)

Let us substitute this in terms of Cot and Cosec as asked for in the problem.

Cos²X = Cot²X/Cosec²X and Sin²X = 1/Cosec²X

Substituting in (1)

Cos(2X) = (Cot²X - 1)/Cosec²X ........ (2)

As we know Cosec²X - Cot²X = 1, substituting in (2) we get

Cos(2X) = (Cot²X - (Cosec²X - Cot²X))/(1 + Cot²X)

= (2Cot²X - Cosec²X)/(1+Cot²X)

Hence proved.

(PS - Note that the question is missing a square symbol in the denominator)