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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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.

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