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

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lbawa | Student, Undergraduate | (Level 1) Honors

Posted April 20, 2011 at 5:55 AM via web

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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

Posted April 22, 2011 at 2:53 AM (Answer #1)

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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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