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Question

Prove that $$\dfrac{{\cos A}}{{1 - \tan A}} + \dfrac{{{\mathop{\rm sinA}\nolimits} }}{{1 - \cot A}} = \cos A + {\mathop{\rm sinA}\nolimits} .$$


Solution

LHS
=cosA/(1-tanA)+sinA/(1-cotA)
=cos A/(1 - sin A/cos A) + sin A/(1 - cos A/sin A)
=cos²A/ (cos A - sin A) + sin²A / (sin A - cos A)
=cos²A/ (cos A - sin A) - sin²A / (cos A - sin A)
=(cos ² A - sin ² A) / (cos A - sin A)
=(cos A - sin A)(cos A + sin A) / (cos A - sin A)
=cos A + sin A i.e RHS

LHS=RHS
Hence, Proved

Maths

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