Question

Prove: (tan A)/((1 + tan^2 A) ^ 2) + (cos A)/((1 + cot^2 A) ^ 2) = sinA * cosA​

Answer 1

Answer:

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Answer 2

$lhs \\ \\ \frac{tan \alpha }{ {(1 + {tan}^{2} \alpha ) }^{2} } + \frac{cos \alpha }{ {(1 + cot { \alpha }^{2}) }^{2} } \\ \\ = \frac{tan \alpha }{ {(1 + ( \frac{sin ^{2} \alpha }{ \cos ^{2} \alpha }) )}^{2} } + \frac{cos \alpha }{ {(1 + ( \frac{cos ^{2} \alpha }{ \sin ^{2} \alpha }) )}^{2} } \\ \\ = \frac{tan \alpha }{ { \frac{ ({cos}^{2} \alpha + {sin}^{2} \alpha )}{ {cos}^{4} \alpha } }^{2} } + \frac{ \cos \alpha }{ \frac{( {sin}^{2} \alpha + {cos}^{2} \alpha ) ^{2} }{ {sin}^{4} \alpha } } \\ \\ = \frac{tan \alpha \: \: {cos}^{4} \alpha }{ {(1)}^{2} } + \frac{ \cos \alpha \: \: {sin}^{4} \alpha }{ {(1)}^{2} } \\ \\ = \tan \alpha \: \: {cos}^{4} \alpha + \cos \alpha \: \: {sin}^{4} \alpha \\ \\ = \frac{sin \alpha }{cos \alpha } \: \: {cos}^{4} \alpha + \cos \alpha \: \: {sin}^{4} \alpha \\ \\ = \sin \alpha \: \: {cos}^{3} \alpha + \cos \alpha \: \: {sin}^{4} \alpha \\ \\ take \: common \\ \\ = sin \alpha \: \: cos \alpha ( {cos}^{2} \alpha + {sin}^{2} \alpha ) \\ \\ = sin \alpha \: \: cos \alpha ( 1 ) \\ \\ = sin \alpha \: \: cos \alpha$

Hence proved...

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