# How do you prove #(tan3t-tant)/(1+tan3t*tant)=(2tant)/(1-tan^2t)#?

Proved

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To prove (tan3t - tan t) / (1 + tan3t * tan t) = (2tan t) / (1 - tan^2 t), we'll start with the left-hand side (LHS) and manipulate it step by step using trigonometric identities.

LHS: = (tan3t - tan t) / (1 + tan3t * tan t) = (tan(2t + t) - tan t) / (1 + tan(2t + t) * tan t) = [(tan2t + tan t) / (1 - tan2t * tan t)] / (1 + tan2t * tan t) = (tan2t + tan t) / (1 - tan^2 t) = 2tan t / (1 - tan^2 t)

This matches the right-hand side (RHS), thus proving the identity.

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