Prove that: `(i)tan^(-1){(sqrt(1+cosx)+sqrt(1-cosx))/(sqrt(1+cosx)-sqrt(1-cosx))}=pi/4+x/2`,

Explanation

`cos 2 theta = 2 cos^2 theta - 1`

`1 + cos 2 theta = 2 cos^2 theta`

so, `1 + cos x = 2cos^2 (x/2)`

Also `cos 2 theta = 1- 2sin^2 theta`

`1- cos 2 theta = 2 sin^2 theta`

so, `1 - cosx = 2sin^2(x/2)`

now, putting it in the equation given

`tan^-1{(sqrt(2cos^2(x/2)) - sqrt(2sin^2(x/2)))/(sqrt(2cos^2(x/2)) - sqrt(2sin^2(x/2)))}`

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