Question

Prove that:

$\frac{ \cos(a) }{1 + \sin(a) } + \frac{1 + \sin(a) }{ \cos(a) } = 2 \sec(a)$
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Answer 1

$\bf\huge{Solution:-}$

To prove :

$\frac{ \cos(a) }{1 + \sin(a) } + \frac{1 + \sin(a) }{ \cos(a) } = 2 \sec(a)$

L.H.S.

$\frac{ \cos(a) }{1 + \sin(a) } + \frac{1 + \sin(a) }{ \cos(a) }$

$= \frac{ { \cos }^{2}(a) + (1 + \sin(a) {)}^{2} }{(1 + \sin(a)) \cos(a) }$

$= \frac{( { \cos }^{2}(a) + { \sin }^{2}(a)) + 1 + 2 \sin(a) }{(1 + \sin(a)) \cos(a) }$

$= \frac{ { \cos }^{2}(a) + { \sin }^{2}(a) + 1 + 2 \sin(a) }{(1 + \sin(a) \cos(a) ) }$

$= \frac{1 + 1 + 2 \sin(a) }{(1 + \sin(a)) \cos(a) } \: \\ therefore \: ( { \sin }^{2} + { \cos }^{2} = 1)$

$= \frac{2 + 2 \sin(a) }{(1 + \sin(a)) \cos(a) }$

$= \: \frac{2(1 + \sin(a)) }{(1 + \sin(a)) \cos(a) }$

$= \frac{2}{ \cos(a) }$

= 2sec(a)=R.H.S.

Therefore, L.H.S. = R.H.S

Hence proved.