SOLUTION: Prove the following is an Identity: Tan (X) - Cot (X) over Tan (X) + Cot (X) = 2sin^2 (X)

SOLUTION: Prove the following is an Identity: Tan (X) - Cot (X) over Tan (X) + Cot (X) = 2sin^2 (X) - 1

Question 35523: Prove the following is an Identity:
Tan (X) - Cot (X) over Tan (X) + Cot (X) = 2sin^2 (X) - 1
Answer by narayaba(40) (Show Source): You can put this solution on YOUR website!
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

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