(cos²A/cosA -sinA) +(sinA/1-cotA) =sinA + cosA​

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Answer:

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[tex]\displaystyle \sf{ \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{sin\:A}{1 - cot\:A} = sin\:A + cos\:A} \: \: is \: \bf proved[/tex]

Given :

[tex]\displaystyle \sf{ \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{sin\:A}{1 - cot\:A} = sin\:A + cos\:A}[/tex]

To find :

To prove the expression

Solution :

Step 1 of 2 :

Write down the given expression to prove

Here the given expression is

[tex]\displaystyle \sf{ \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{sin\:A}{1 - cot\:A} = sin\:A + cos\:A}[/tex]

Step 2 of 2 :

Prove the expression

LHS

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{sin\:A}{1 - cot\:A} }[/tex]

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{sin\:A}{1 - \dfrac{cos\:A}{sin\:A} } }[/tex]

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{sin\:A}{ \dfrac{sin\:A - cos\:A}{sin\:A} } }[/tex]

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{{sin}^{2} \:A}{sin\:A - cos\:A} }[/tex]

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} + \frac{{sin}^{2} \:A}{ - (cos\:A - sin \:A)} }[/tex]

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A }{ cos\:A - sin \:A} - \frac{{sin}^{2} \:A}{ cos\:A - sin \:A} }[/tex]

[tex]\displaystyle \sf{ = \frac{{cos}^{2} \:A - {sin}^{2} \:A}{ cos\:A - sin \:A} }[/tex]

[tex]\displaystyle \sf{ = \frac{(cos\:A + sin \:A)(cos\:A - sin \:A)}{ cos\:A - sin \:A} }[/tex]

[tex]\displaystyle \sf{ = (sin\:A + cos \:A) }[/tex]

= RHS

Hence the proof follows

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