Question

prove that cos theta /1-tan theta + sin theta /1-cot theta =sin theta +cos theta

Answer 1

$\small{ =>\frac{ \cos A }{1 - \tan A } + \frac{ \sin A }{1 - \cot A} } \\ \\ \\ \small{ => \frac{ \cos A }{1 - \frac{ \sin A }{ \cos A } } + \frac{ \sin A }{1 - \frac{ \cos A}{ \sin A } }} \\ \\ \\ \small{=> \frac{ \cos A }{ \frac{ \cos A - \sin A }{ \cos A} } + \frac{ \sin A }{ \frac{ \sin A - \cos A}{ \sin A } }} \\ \\ \\ \small{ => \frac{ \cos ^{2} A }{ \cos A - \sin A} + \frac{ \sin^{2} A }{ \sin A - \cos A} } \\ \\ \\ \small{ => \bold{ - } \frac{ \cos ^{2} A}{ \sin A - \cos A } + \frac{ \sin^{2} A }{ \sin A - \cos A} } \\ \\ \\ \small{ => \frac{ \sin ^{2} A- \cos^{2} A }{ \sin A - \cos A }} \\ \\ \\ \small{=> \frac{( \sin A - \cos A )( \sin A + \cos A )}{ \sin A - \cos A }} \\ \\ \\ \small{ => \sin A + \cos A}$

Hence, proved.

Answer 2

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