# Prove the following identities: tan3θ1+tan2θ+cot3θ1+cot3θ = secθ cosecθ – 2sinθ cosθ - Mathematics and Statistics

Sum

Prove the following identities:

tan^3theta/(1 + tan^2theta) + cot^3theta/(1 + cot^3theta = secθ cosecθ – 2sinθ cosθ

#### Solution

L.H.S> = tan^3theta/(1 + tan^2theta) + cot^3theta/(1 + cot^2theta

= tan^3theta/sec^2theta + cot^3theta/("cosec"^2theta)

= (((sin^3theta)/cos^3 theta))/((1/cos^2theta)) + (((cos^3 theta)/(sin^3 theta)))/((1/sin^2 theta)

= sin^3theta/costheta + cos^3theta/sintheta

= (sin^4theta + cos^4 theta)/(sintheta cos theta)

= ((sin^2 theta)^2 + (cos^2 theta)^2)/(sin theta cos theta

= ((sin^2 theta + cos^2 theta)^2 - 2sin^2 theta cos^2 theta)/(sintheta cos theta)   ...[∵ a2  + b2 = (a + b)2 - 2ab]

= (1^2 - 2sin^2 theta cos^2 theta)/(sin theta cos theta)

= 1/(costheta*sintheta) - (2sin^2theta cos^2 theta)/(sintheta cos theta)

= secθ cosecθ – 2sinθ cosθ

= R.H.S.

Concept: Fundamental Identities
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