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Question

Prove that :
$$\dfrac{cos \, A}{1 \, - \, sin \, A} \, + \, \dfrac{sin \, A}{1 \, - \, cos \, A} \, + \, 1 \, = \, \dfrac{sin \, A \, cos \, A}{(1 \, - \, sin \, A)(1 \, - \, cos \, A)}$$


Solution

we have
$$ \Rightarrow  \, \, \, LHS \, = \, \dfrac { cos \, A}{1 \, - \, sin A} \, + \, \dfrac { sin \, A}{1 \, - \, cos A}  \, + \,1 $$

$$ \Rightarrow  \, \, \, LHS \, = \dfrac { cos A (1 \, - \, cos A ) \, + \,  sin A (1 \, - \, cos A ) \, + \, (1 \, - \, sin A )(1 \, + \, cos A) }{  sin A (1 \, - \, cos A ) \, + \, (1 \, - \, sin A )(1 \, + \, cos A) }$$

$$ \Rightarrow  \, \, \, LHS \, = \dfrac { cos A  \, - \, cos^2 A  \, + \, sin A \, - \, sin ^2 A  \, + \, 1 \, - \, sin A \, - \, cos A \, + \, sin A cos A}{ sin A (1 \, - \, cos A ) \, + \, (1 \, - \, sin A )(1 \, + \, cos A) }$$

$$ \Rightarrow  \, \, \, LHS \, = \, \dfrac {(cos A \, + \, sin A) \, -\, (cos^2 A \, - \, sin ^2A ) + \, 1 \, - \, ( cos A \, + , sin A)  \, + \, sin A cos A  }{ sin A (1 \, - \, cos A ) \, + \, (1 \, - \, sin A )(1 \, + \, cos A) }$$

$$ \Rightarrow  \, \, \, LHS \, = \, \dfrac{ (cos A \, + \, sin \, A) - \, 1 \, + \,1 \, - \, ( cos A \, + \, sin A)  \, + \, sin A \, + \, cos A}{ sin A (1 \, - \, cos A ) \, + \, (1 \, - \, sin A )(1 \, + \, cos A) } $$

$$ \Rightarrow  \, \, \, LHS \, = \dfrac {sin A \, cos A}{ sin A (1 \, - \, cos A ) \, + \, (1 \, - \, sin A )(1 \, + \, cos A) } $$ $$= RHS$$ 

Mathematics

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