Prove the following: `cot^(-1)[(sqrt(1+sinx )+sqrt(1-sinx))/(sqrt(1+sinx)-\ sqrt(1-sinx))]=x/2,\ x (0,pi/4)`

Explanation

`sqrt(1+sinx) = sqrt(1+2sin(x/2)cos(x/2))` (As `sin2theta = 2sinthetacostheta`)

`=sqrt(cos^2(x/2)+sin^2(x/2)+2sin(x/2)cos(x/2))` (As `cos^2theta+sin^2theta = 1`)

`=sqrt(cos(x/2)+sin(x/2)^2)`

`:. sqrt(1+sinx) =cos(x/2)+sin(x/2)`

Now, `sqrt(1-sinx) = sqrt(1-2sin(x/2)cos(x/2))` (As `sin2theta = 2sinthetacostheta`)

`=sqrt(cos^2(x/2)+sin^2(x/2)-2sin(x/2)cos(x/2))`

`=sqrt(cos(x/2)-sin(x/2)^2)`

`:.sqrt(1-sinx) =cos(x/2)-sin(x/2)`

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