Prove the following trigonometric identities: tanA/(1+tan^2A)^2+cotA/(1+cot^2A)^2 = sinAcosA

Question

Prove the following trigonometric identities:

\(\frac{tanA}{(1+tan^2A)^2}+\frac{cotA}{(1+cot^2A)^2}\) = sinA cosA

Answer 1

\(\frac{tanA}{(1+tan^2A)^2}+\frac{cotA}{(1+cot^2A)^2}\) = \(\frac{tanA}{(sec^2A)^2}+\frac{cotA}{(cosec^2A)^2}\) 

\(\frac{sinA}{cosA}\times cos^4A+\frac{cosA}{sinA}\times sin^4A\)

= sinA cos³A + cosAsin³A

= sinA cosA + (cos²A + sin²A)

= sinA cosA x 1

= sinA cosA

Hence Proved