Question

What is the interval of the following function for which,`2tan^(-1)x+sin^(-1)[(2x)/(1+x^2)]` is independent of `x`?

Answer 1

`x^2 ge 1`

Explanation:

#sin(2tan^(-1)x+sin^(-1)((2x)/(1+x^2))) = (2 x)/(1 + x^2)^2 - (2 x^3)/(1 + x^2)^2 + ( 2 x sqrt[1 - (4 x^2)/(1 + x^2)^2])/(1 + x^2)=#

`= 2x(1-x^2+abs(x^2-1))/(x^2+1)^2`

then it is independent of `x` for

`x^2-1 ge 0`

because then

`2x(1-x^2+abs(x^2-1))/(x^2+1)^2 = 0`