Verify the identity arcsin x=[arcsin(3x-4x^3)]/3

To verify if the given identity is holding, we'll shift
all terms to one side and we'll assign a function f(x) to the given
expression.

f(x) = 3arcsin x -[arcsin(3x-4x^3)] =
0

We'll use one of Lagrange's consequences. If the
derivative of the function f(x) is cancelling out, then the function is a
constant.

We'll calculate the first derivative of the
function, both sides:

{3arcsin x -[arcsin(3x-4x^3)]}' =
(0)'

(arcsin x)' =
1/sqrt(1-x^2)

[arcsin(3x-4x^3)]' = (3x-4x^3)'/sqrt[1 -
(3x-4x^3)^2]

[arcsin(3x-4x^3)]' = (3 - 12x^2)/sqrt[1 -
(3x-4x^3)^2] (2)

(3arcsinx)' = 3/sqrt(1-x^2)
(1)

{3arcsin x -[arcsin(3x-4x^3)]}' = (1) -
(2)

3/sqrt(1-x^2) - (3 - 12x^2)/sqrt[1 - (3x-4x^3)^2]
=

If (1) - (2) = 0, then the identity is
verified.

Another method is to calculate sine function both
sides:

sin (3arcsin x) = sin
[arcsin(3x-4x^3)]

We'll note arcsin x = t => sin t =
x and we'll use the identity sin (arcsin x) = x.

sin 3t =
(3x-4x^3)

We'll use the triple angle
identity;

sin 3t = sin (2t+t) = sin2t*cost +
sint*cos2t

sin3t = 2sint*[1-(sin t)^2] + sint*[1-2(sin
t)^2]

sin3t = 2sint - 2(sin t)^3 + sint - 2(sin
t)^3

sin3t = 3sint- 4(sin
t)^3

We'll replace sin t by
x:

sin3t = 3x -
4x^3

We notice that LHS = RHS, therefore the
identity arcsin x = [arcsin(3x-4x^3)]/3 is
verified.