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Prove that: (tan A)/(1+tan^2A)^2 + (cot...

Prove that: `(tan A)/(1+tan^2A)^2 + (cot A)/(1+cot^2A)^2 = sin A cos A `.

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LHS = `(tan A)/(1+tan^2A)^2 +(cot A)/(1+cot^2A)^2`
`(tan A)/((sec^2 A)^2) + cot/((cosec^2 A)^2)`
…`[(therefore sec^2 A = 1-tan^2A)/(cosec^2A=1 + cot^2A)]`
=`(tan A)/(sec^4 A) + cot/(cosec^4 A)`
= `(tan A div sec^4 A) + (cot div cosec^4 A)`
`=(sinA/cosA div 1/cos^4A) + (cosA/sinA div 1/sin^4A)`
`....[ tan A = sinA/cosA, sec A = 1/(cos A)`,
`cotA = cosA/sinA,cosecA = 1/sinA`]
`((sinA)/(cosA)xx(cos^4A)/(1)) + (cosA/sinA xx sin^4 A)`
`=sinA xx cos^3A +cos A xx sin^3 A`
`=sin A xx cos A(cos^2A + sin^2A)`
=`sin A xx cos A (1)...(therefore sin^2A + cos^2A =1)`
` = sin A xx cosA `
=RHS
`therefore`LHS = RHS
`therefore (tan A)/(1+tan^2A)^2 +(cot A)/(1+cot^2A)^2 = sin A xx cos A`.
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