# Prove the identity: (tan^2A)/(1+tan^2A) + (cot^2A)/(1+cot^2A)=(1-2sin^2A cos^2A)/(sinAcosA)

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The identity that has to be proved is:

(tan A)^2/(1+ (tan A)^2) + (cot A)^2/(1+(cot A)^2) = (1- 2(sin A)^2 (cos A)^2)/(sin A)(cos A)

Starting with the left hand side:

(tan A)^2/(1+ (tan A)^2) + (cot A)^2/(1+(cot A)^2)

use cot A = 1/(tan A)

=> (tan A)^2/(1+ (tan A)^2) + (1/tan A)^2/(1+(1/tan A)^2)

=> (tan A)^2/(1+ (tan A)^2) + (1/tan A)^2/[(1 + (tan A)^2)/(tan A)^2)]

=> (tan A)^2/(1+ (tan A)^2) + (tan A)^2)(1/tan A)^2/(1 + (tan A)^2)

=> (tan A)^2/(1+ (tan A)^2) + 1/(1 + (tan A)^2)

=> ((tan A)^2 + 1)/(1+ (tan A)^2)

=> 1

Now the right hand side

(1- 2(sin A)^2 (cos A)^2)/(sin A)(cos A)

=> 1/(sin A)(cos A) - 2*(sin A)(cos A)

The right hand side does not equal 1. So the left hand side and the right hand side are not equal.

**The given expression is not an identity.**