Write `cot^(-1) (sqrt(1+x^(2))-x)` in the simplest form.

Express tan^(-1) ((3a^(2)x -x^(3))/( a^(3) -3ax^(2))) , where a gt 0, (-a)/(sqrt(3)) le x le (a)/(sqrt(3)) in the simplest form.

Express tan^(-1)(sqrt((1-cosx)/(1+cosx))), x lt pi , in the simplest form.

Prove that cot^(-1) ((sqrt(1+sin x) +sqrt(1-sin x))/(sqrt(1+sin x) -sqrt(1-sinx)))=(x)/(2), x in (0, (pi)/(4)) .

If "tan"^(-1) (sqrt(1+x^(2))-sqrt(1-x^(2)))/(sqrt(1+x^(2))+sqrt(1-x^(2)))=alpha , then prove that x^(2) =sin 2alpha .

Find the value of cos^(-1)x +cos^(-1) sqrt(1-x^(2)) .