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

1 Answer | Add Yours

justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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.

We’ve answered 333,802 questions. We can answer yours, too.

Ask a question