Question

Prove the following trigonometric identities:
tan3θ/1+tan²θ+cot3θ/1+cot²θ=secθcosecθ-2sinθcosθ

Answer 1

Answer:

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Answer 2

tan³θ /(1+tan²θ) + cot³θ / (1+cot²θ) = secθ. cosecθ - 2sinθ. cosθ  proved

Step-by-step explanation:

To prove: tan³θ /(1+tan²θ) + cot³θ / (1+cot²θ) = secθ. cosecθ - 2sinθ. cosθ

Proof:  

LHS tan³θ /(1+tan²θ) + cot³θ / (1+cot²θ) = [Sin³θ/Cos³θ / (1 + Cos²θ/Sin²θ)]

= [Sin³θ/Cos³θ / (Cos²θ + Sin²θ)/Cos²θ)] + [Cos³θ/Sin³θ/ (Cos²θ + Sin²θ)/Sin²θ)]

= Sin³θ/Cosθ + Cos³θ/Sinθ

= Sin^4θ + Cos^4θ / Cosθ.Sinθ

= [ 1- 2Sin²θCos²θ ] / Cosθ.Sinθ  -----{from a²+b² = (a+b)² - 2ab}

= 1/Cosθ.Sinθ - 2.Sinθ.Cosθ

= Secθ.Cosecθ - 2.Sinθ.Cosθ

RHS = LHS

Hence proved.