Question

tan^3/1+tan^2+cot^3/1+cot^2=sec.cosec -2sin.cos​

Answer 1

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LHS = Tan^3A / ( 1+ Tan^2A) + Cot^3A / (1 + Cot^2a) 

= Tan^3A / Sec^2A + Cot^3A / Cosec^2A 

= (sin^3A/cos^3A) / (1 / Cos^2A) + (Cos^3A/Sin^3A) / (1 / Sin^2A) 

= Sin^3A/CosA + Cos^3A/SinA 

= (Sin^4A + Cos^4A) / SinA.CosA 

= [ (Sin^2A + Cos^2A)^2 - 2Sin^2A.Cos^2A] / SinA.CosA 

= ( 1- 2Sin^A.Cos^A)/ SinA.CosA 

RHS = SecA CosecA - 2sinAcosA 

= 1/CosA . 1/SinA - 2SinACosA 

= (1 - Sin^2A.Cos^2A) / sinAcosA 

Hence LHS = RHS (PROVED)