Prove \small ( \sin(A) - \cos(A) + 1)/(\sin(A) + \cos(A) - 1) = (\cos(A))/(1 - \sin(A) ) Help!!!!!!!!!

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asked Aug 1, 2023 88.5k views

1 Answer

AppleLover · Answer 1 · 2023-08-07T04:08:47+0000

$( \sin(a) - \cos(a) + 1 )/( \sin(a) + \cos(a) - 1 ) = \\$

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$( \sin(a) - \cos(a) + 1 )/( \sin(a) + \cos(a) - 1 ) * ( \sin(a) + \cos(a) + 1)/( \sin(a) + \cos(a) + 1 ) =$

$\frac{ {sin}^(2)(a) + 2 \sin(a) - {cos}^(2) (a) + 1 }{ {sin}^(2)(a) + 2 \sin(a) \cos(a) + {cos}^(2)(a) - 1 } =$

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As you know :

${sin}^(2) (a) + {cos}^(2) (a) = 1$

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$\frac{ {sin}^(2) (a) - {cos}^(2)(a) + 2 \sin(a) + 1}{ {sin}^(2) (a) + {cos}^(2)(a) - 1 + 2 \sin(a) \cos(a) } =$

$\frac{ {sin}^(2)(a) - (1 - {sin}^(2)(a)) + 2 \sin(a) + 1 }{1 - 1 + 2 \sin(a) \cos(a) } =$

$\frac{ {sin}^(2) (a) + {sin}^(2) (a) - 1 + 1 + 2 \sin(a) }{2 \sin(a) \cos(a) } =$

$\frac{2 {sin}^(2)(a) + 2 \sin(a) }{2 \sin(a) \cos(a) } =$

$(2 \sin(a)( \sin(a) + 1) )/(2 \sin(a)( \cos(a) \: ) ) = \\$

$( \sin(a) + 1)/( \cos(a) ) \\$

And we're done...

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