Question

sin x ( 1+tan x) +cos x (1+cotx)=sec x +cosec x


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Answer 1

Solution:-

On taking LHS

$= \sin(x) (1 + \tan(x) ) + \cos(x) (1 + \cot(x) ) \\ \\ = \sin(x) + \frac{ { \sin}^{2}x }{ \cos(x) } + \cos(x) + \frac{ \cos {}^{2} ( x) }{ \sin(x) } \\ \\ = \frac{ \sin {}^{2} (x) \cos(x) + \sin {}^{3} (x) + \sin(x) \cos {}^{2} (x) + \cos {}^{3} (x) }{ \sin(x) \cos(x) } \\ \\ = \frac{\sin {}^{2} (x) \cos(x) + \cos {}^{3} (x) + \sin {}^{3} (x) + \sin(x) \cos {}^{2} (x)}{ \sin(x) \cos(x) } \\ \\ = \frac{ \cos(x) ( \sin {}^{2} (x) + \cos {}^{2} (x)) + \sin(x)( \sin {}^{2} (x) + \cos {}^{2} (x)) }{ \cos(x) \sin(x) } \\ \\ = \frac{ \cos(x) + \sin(x) }{ \cos(x) \sin(x) } \\ \\ = \frac{1}{ \sin(x) } + \frac{1}{ \cos(x) } \\ \\ = \csc(x) + \sec(x) \\ \\ = \sec(x) + \csc(x)$

= RHS

HENCE PROVED.

Answer 2

Answer:

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