Wednesday, May 13, 2015

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

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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