Question 35523
Tan (X) - Cot (X) / Tan (X) + Cot (X)-------------------(1)

Cot (X) = 1/Tan (X)

pluggin in for Cot(X) in (1)

(Tan(X) - 1/Tan(X)) / (Tan(X) + 1/Tan(X))

=((Tan^2(X) - 1)/Tan(X)) / ((Tan^2(X) + 1)/Tan (X))

=(Tan^2(X) - 1)/ (Tan^2(X) + 1) 

=(Tan^2(X) - 1)/ Sec^2(X)

=cos^2(X) * (Tan^2(X) - 1)

=cos^2(X) * ((sin^2(X)/cos^2(X)) - 1)

=sin^2(X) - cos^2(X) = sin^2(X) - (1- sin^2(X)) = 2sin^2(X) - 1