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

Can someone help me prove 8 a,b, and c-example-1

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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

User Yuri Baburov
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