Take the tangent of both sides,

tan(arctan(x))=tan(arccos(2x))

x=tan(arccos(2x))

Let,

arccos(2x)=u

Then,

cos u =2x=(2x)/1

Thus,

A right triangle angle is "u" with adjacent side 2x and hypotenuse 1.

By Pythagorean theorem the opposite side is sqrt(1-4x^2)

Then,

tan u

Is opposite over adjacent which is,

sqrt(1-4x^2)/(2x)

Thus we finally have,

x=sqrt(1-4x^2)/(2x)

Multiply by 2x,

2x=sqrt(1-4x^2)

Square,

4x^2=1-4x^2

Thus,

8x^2=1

Thus,

x^2=1/8