`tan^(-1)[(cosx)/(1+sinx)]` is equal to `pi/4-x/2,forx in (-pi/2,(3pi)/2)` `pi/4-x/2,forx in (-pi/2,pi/2)` `pi/4-x/2,forx in (-pi/2,pi/2)` `pi/4-x/2,forx in (-(3pi)/2,pi/2)`

Explanation

`tan^-1((cosx)/(1+sinx))`

`=tan^-1((sin(pi/2-x))/(1+cos(pi/2-x)))`

`=tan^-1((2sin(pi/4-x/2)cos(pi/4-x/2))/(2cos^2(pi/4-x/2)))`

`=tan^-1(tan(pi/4-x/2)`

`=pi/4-x/2`

Now, we know, range of `tan^-1x` is from `-pi/2` to `pi/2`.

`:. -pi/2 lt pi/4-x/2 lt pi/2`

`=> -(3pi)/4 lt -x/2 lt pi/4`

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