Evaluate 1 + sin x /1-sin x

We need to evaluate 1+sin /1-sinx

Solution

1+sin /1-sinx

To the given equation we will multiply and divide by [ 1 + sinx]

So we get the equation as

(1 + sinx) (1 + sinx) / (1 – sinx) (1 + sinx)

(1 – sinx) (1 + sinx) can be expressed as (1 – sin2x) using the identity (ab) (a+b) = (a2-b2)

=(1 + sinx)2 / (1 – sin2x)

We know that sin2x + cos2x = 1 so 1 – cos2x= sin2x

=(1 + sinx)2 / cos2x

= { (1 + sinx) / cos x} 2

= {sec x + tan x }2

Answer

1+sin /1-sinx ={sec x + tan x }2

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