Question

prove that cosA/1+tanA - sinA/1+cotA = cosA-sinA

Answer 1

$\frac{cos \alpha }{1 + \tan \alpha } - \frac{sin \alpha }{ 1 + \cot \alpha } \\ = \cos \alpha - \sin \alpha \\ \\ LHS \\ \\ \frac{ \cos \alpha }{1 + \frac{sin \alpha }{ \cos \alpha } } - \frac{ \sin \alpha }{1 + \frac{ \cos \alpha }{ \sin\alpha } } \\ \\ = \frac{cos \alpha \times \cos \alpha }{ \sin \alpha + \cos \alpha } - \frac{ \sin\alpha \times \sin \alpha }{ \sin \alpha + \cos \alpha } \\ \\ = \frac{ { \cos }^{2} \alpha - { \sin}^{2} \alpha }{ \sin \alpha + \cos \alpha } \\ \\ = \frac{( \cos \alpha + \sin \alpha)( \cos \alpha - \sin \alpha)} { \sin \alpha + \cos \alpha } \\ \\ = \cos\alpha - \sin\alpha \\ \\ \huge \: Hence \: Proved$