Question

Can someone help me prove 8 a,b, and c

Answer 1

Step-by-step explanation:

8) Consider,

$\frac{1}{1 + \sin(x) }$

Multiplying And Dividing By 1-sinx

$\frac{1}{1 + \sin(x) } \times \frac{1 - \sin(x) }{1 - \sin(x) }$

$\frac{1 - \sin(x) }{(1 + \sin(x) )(1 - \sin(x)) }$

We know (a+b)(a-b)=a²-b²

$\frac{1 - \sin(x) }{1 - \sin ^{2} (x) }$

We know that sin²x+cos²x=1

cos²x=1-sin²x,

uisng this we get,

$\frac{1 - \sin(x) }{ \cos ^{2} (x) }$

$\frac{1}{ \cos ^{2} (x) } - \frac{ \sin(x) }{cos ^{2} (x)}$

We know that 1/cosx=secx and sinx/cosx=tanx

Using this we get,

$\sec {}^{2} (x) - \frac{ \tan(x) }{ \cos(x) }$

Hence Proved