Question

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

Answer 1

Explanation:

8) Consider,

`(1)/(1 + \sin(x) )`

Multiplying And Dividing By 1-sinx

`(1)/(1 + \sin(x) ) * (1 - \sin(x) )/(1 - \sin(x) )`

`(1 - \sin(x) )/((1 + \sin(x) )(1 - \sin(x)) )`

We know (a+b)(a-b)=a²-b²

`(1 - \sin(x) )/(1 - \sin ^(2) (x) )`

We know that sin²x+cos²x=1

cos²x=1-sin²x,

uisng this we get,

`(1 - \sin(x) )/( \cos ^(2) (x) )`

`(1)/( \cos ^(2) (x) ) - ( \sin(x) )/(cos ^(2) (x))`

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

Using this we get,

`\sec {}^(2) (x) - ( \tan(x) )/( \cos(x) )`

Hence Proved