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Prove: ^{ 2 }{ A( { { A-1 } }{ 1+ { A } } ) } + ^{ 2 }{

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Prove:
$cot^{2} A (\frac{sec A - 1}{1 + sin A}) + sec^{2} A (\frac{sin A - 1}{1 + sec A}) = 0$

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Prove:

cot^{2} A (\frac{sec A - 1}{1 + sin A}) + sec^{2} A (\frac{sin A - 1}{1 + sec A}) = 0

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Text solutionVerified

co t^{2} A (\frac{sec A - 1}{1 + s in A}) + sec^{2} A (\frac{s in A - 1}{1 + sec A})

Rationalising the above L.H.S,

⟹ co t^{2} A (\frac{( sec A - 1 ) ( 1 - s in A )}{( 1 + s in A ) ( 1 - s in A )}) + sec^{2} A (\frac{( s in A - 1 ) ( 1 - sec A )}{( 1 + sec A ) ( 1 - sec A )})

⟹ co t^{2} A (\frac{( sec A - sec A s in A - 1 + s in A )}{( 1 - s in ^{2} A )}) + sec^{2} A (\frac{( sec A - sec A s in A - 1 + s in A )}{( 1 - sec ^{2} A )})

⟹ \frac{cos ^{2} A}{s in ^{2} A} (\frac{( sec A - sec A s in A - 1 + s in A )}{( cos ^{2} A )}) + \frac{1}{cos ^{2} A} (\frac{( sec A - sec A s in A - 1 + s in A )}{( t an ^{2} A )})

Since,

1 - s in^{2} A = cos^{2} A

⟹ \frac{( sec A - sec A s in A - 1 + s in A )}{( s in ^{2} A )} + \frac{( sec A - sec A s in A - 1 + s in A )}{( s in ^{2} A )}

⟹ \frac{( sec A - sec A s in A - 1 + s in A + sec A - sec A s in A - 1 + s in A )}{( s in ^{2} A )}

⟹ \frac{0}{s in ^{2} A}

⟹ 0

⟹ R . H . S

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Question Text	Prove: $cot^{2} A (\frac{sec A - 1}{1 + sin A}) + sec^{2} A (\frac{sin A - 1}{1 + sec A}) = 0$
Answer Type	Text solution:1
Upvotes	150