`sin^(2)(2tan^(-1)sqrt((1+x)/(1-x)));|x|<1=`

sin^(2)(2tan^(-1)sqrt((1+x)/(1-x)))="____","where "-1lexlt1

Write each of the following in the simplest form: (i) sin^(-1){(sqrt(1+x)+sqrt(1-x))/2},\ \ 0 < x <1 (ii) sin{2tan^(-1)(sqrt((1-x)/(1+x)))}

(iv) If y=tan^(-1)(x/(1+sqrt(1-x^(2))))+sin(2tan^(-1)sqrt((1-x)/(1+x))) , then find (dy)/(dx) for x epsilon(-1,1)

Differentiate tan^(-1)(x)/(1+sqrt((1-x^(2))))+{2tan^(-1)sqrt(((1-x)/(1+x)))}sin w.r.t.x

Simplify :sin[2tan^(-1)sqrt((1-x)/(1+x))]

Prove that sin [2 tan^(-1) {sqrt((1 -x)/(1 + x))}] = sqrt(1 - x^(2))

Find (dy)/(dx) when (y-tan^(-1))(x)/(1+sqrt(1-x^(2)))+sin[2tan^(-1)sqrt((1-x)/(1+x))]

y=tan^(-1)((x)/(1+sqrt(1-x^(2))))+sin(2tan^(-1)theta*sqrt((1-x)/(1+x))) then prove that ,4(1-x^(2))^(3)((d^(2)y)/(dx^(2)))^(2)+4x=x^(2)+4