Du hast doch dann bereits alles gegeben was zur Lösung notwendig ist.
ARCTAN(2·x) = 2·ARCCOS(x)
ARCSIN((2·x)/√(1 + (2·x)^2)) = 2·ARCCOS(x)
ARCSIN(x) + ARCCOS(x) = pi/2 --> ARCCOS(x) = pi/2 - ARCSIN(x)
ARCSIN((2·x)/√(1 + (2·x)^2)) = 2·(pi/2 - ARCSIN(x))
ARCSIN(2·x/√(4·x^2 + 1)) = pi - 2·ARCSIN(x)
SIN(ARCSIN(2·x/√(4·x^2 + 1))) = SIN(pi - 2·ARCSIN(x))
SIN(x ± y) = SIN(x)·COS(y) ± COS(x)·SIN(y)
2·x/√(4·x^2 + 1) = SIN(pi)·COS(2·ARCSIN(x)) - COS(pi)·SIN(2·ARCSIN(x))
2·x/√(4·x^2 + 1) = 0 - (-1)·SIN(2·ARCSIN(x))
2·x/√(4·x^2 + 1) = SIN(2·ARCSIN(x))
SIN(2·x) = SIN(x + x) = SIN(x)·COS(x) + COS(x)·SIN(x) = 2·SIN(x)·COS(x)
2·x/√(4·x^2 + 1) = 2·SIN(ARCSIN(x))·COS(ARCSIN(x))
2·x/√(4·x^2 + 1) = 2·x·√(1 - x^2) --> x = 0
1/√(4·x^2 + 1) = √(1 - x^2)
√(1 - x^2)·√(4·x^2 + 1) = 1
(1 - x^2)·(4·x^2 + 1) = 1
- 4·x^4 + 3·x^2 + 1 = 1
- 4·x^4 + 3·x^2 = 0
4·x^4 - 3·x^2 = 0
x^2·(4·x^2 - 3) = 0
x = 0 oder x = ± √(3/4) = ± √3/2