Question

(cos a÷ 1-tan a) + (sin a ÷ 1- cot a) = cos a + sin a​

Answer 1

Answer:

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Answer 2

Question :

Prove that (cos a÷ 1-tan a) + (sin a ÷ 1- cot a) = cos a + sin a.

Solution:

We want to prove that

$\frac{ \cos(a) }{1 - \tan(a) } + \frac{ \sin(a) }{1 - \cot(a) } = \sin(a) + \cos(a)$

LHS

$\frac{ \cos(a) }{1 - tan(a)} + \frac{ \sin(a) }{1 - \cot(a) } \\ = \frac{ \cos(a) }{1 - \frac{sin(a)}{ \cos(a) } } + \frac{sin(a)}{1 - \frac{ \cos(a) }{ \sin(a) } } \\ = \frac{ \cos(a) }{ \frac{ \cos(a) - \sin(a) }{ \cos(a) } } + \frac{sina}{ \frac{ \ \sin(a) - \cos(a) }{ \sin(a) } } \\ = \frac{ { \cos(a) }^{2} }{ \cos(a) - \sin(a) } + \frac{ { \sin(a) }^{2} }{ \sin(a) - \cos(a) } \\ = \frac{ { \cos(a) }^{2} - { \sin(a) }^{2} }{ \cos(a) - \sin(a) } \\ = \frac{( \cos(a) + \sin(a)) \: ( \cos(a) - \sin(a)) }{ \cos(a) - \sin(a) } \\ = \cos(a) + \sin(a)$

Hence, proved.

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