Prove the trigonometric identity 1 cosx1 cosx 2csc2 x 2

Prove the trigonometric identity. 1 - cos(x)/1 + cos(x) = 2csc^2 (x) - 2 csc(x) cot(x) - 1

Solution

We can derive LHS from RHS and prove the identity.

RHS = 2csc²x - 2csc(x)cot(x) -1

       = 2/sin²x - 2cosx/sin²x - 1

       = 2-2cosx - sin²x/sin²x

       = (2 - 2cosx -(1-cos²x))/sin²x (using indentity cos²x + sin²x = 1)

       = cos²x - 2cosx + 1/ sin²x

      = (1-cosx)²/sin²x

      = (1-cosx)²/(1-cos²x)

      = (1-cosx)²/(1+ cosx)(1- cosx)

      = (1-cosx)/(1+cosx) = LHS

Hence proved!!