Prove that tan A/(1 + tan^2 A)^2 + Cot A/(1 + Cot^2 A)^2 = sin A cos A.

Solution

We can start with the left-hand side of the equation and try to simplify it using trigonometric identities:

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

= (sin A/cos A)/(1 + (sin A/cos A)^2)^2 + (cos A/sin A)/(1 + (cos A/sin A)^2)^2 (Using tanA = sinA/cosA and cotA = cosA/sinA)

= (sin A/cos A)/[(cos^2 A + sin^2 A)/(cos^2 A)^2]^2 + (cos A/sin A)/[(sin^2 A + cos^2 A)/(sin^2 A)^2]^2 (using the identity (a/b)^2 = (a^2)/(b^2))

= (sin A/cos A)/[(1)/(cos^4 A)] + (cos A/sin A)/[(1)/(sin^4 A)] (using the identity cos^2 A + sin^2 A = 1)

= sin A cos^3 A + cos A sin^3 A

= sin A cos A (sin^2 A + cos^2 A)

= sin A cos A

Therefore, we have shown that tan A/(1 + tan^2 A)^2 + Cot A/(1 + Cot^2 A)^2 = sin A cos A using trigonometric identities.