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Question
Prove the following trigonometric identities.
`tan A/(1 + tan^2 A)^2 + cot A/((1 + cot^2 A)) = sin A cos A`
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Solution
We have to prove `tan A/(1 + tan^2 A)^2 + cot A/((1 + cot^2 A)) = sin A cos A`
We know that `sin^2 A + cos^2 A = 1`
So
`tan A/(1 + tan^2 A)^2 + cot A/(1 + cot^2 A)^2`
`= tan A/(sec^2 A)^2 + cot A/(cosec^2 A)^2`
`= tan A/sec^4 A + cot A/(cosec^4 A)`
`= (sin A/cos A)/(1/cos^4 A) + (cos A/sin A)/(1/sin^4 A)`
`= (sin A cos^4 A)/cos A + (cos A sin^4 A)/sin A`
`= sin A cos^3 A + cos A sin^3 A`
`= sin A cos A (cos^2 A + sin^2 A)`
= sin A cos A
Hence proved.
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