Question

prove that cosA/1+sinA +1+sinA/cosA=2SecA​

Answer 1

Step-by-step explanation:

We have to prove that,

$\frac{cosA}{1+sinA}+\frac{1+sinA}{cosA}=2secA$

Proof:

For simplicity,

Let's denote angle A as alpha.

Therefore,

$LHS = \frac{ \cos( \alpha ) }{1 + \sin( \alpha ) } + \frac{1 + \sin( \alpha ) }{ \cos( \alpha ) }$

Taking LCM and solving,

we get,

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

But,

we know that,

${ \sin }^{2} \alpha + { \cos }^{2} \alpha = 1$

Therefore,

Putting the values,

we get,

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

But,

we know that,

$\frac{1}{ \cos( \alpha ) } = \sec( \alpha )$

Therefore,

putting the values,

we get,

$= 2 \sec \alpha = RHS$

Therefore,

$\bold{\frac{cosA}{1+sinA}+\frac{1+sinA}{cosA}=2secA}$

Hence, Proved