If `y=tan^-1frac{sqrt(1+x^2)+sqrt(1-x^2)}{sqrt(1+x^2)-sqrt(1-x^2)]`, show that `x^2=sin 2y`

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 .

Show that tan^-1(frac{sqrt(1+x)-sqrt(1-x)}{sqrt(1+x)+sqrt(1-x)})=pi/4-1/2cos^-1 x

Find the derivate of : y= tan^-1(frac(sqrt(1+sinx)- sqrt(1 - sin x))(sqrt(1 +sinx)+sqrt(1-sin x)))

Show that : cot^-1{frac{sqrt(1+sinx)+sqrt(1-sinx)}{sqrt(1+sinx)-sqrt(1-sinx)}}=x/2

Prove that tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))=(pi)/(4)+(1)/(2) cos^(-1)x^(2) .

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 y=sqrt(x)+(1)/sqrt(x) , then show that 2x(dy)/(dx)+y=2sqrt(x) .