Prove that arcsin(3x-4x^3)-3arcsinx=0

We have to prove that
arcsin(3x-4x^3)-3arcsinx=0

Here we first take a
function

f(x) = arcsin(3x-4x^3) -
3arcsinx

and as it is given that arcsin(3x-4x^3) -
3arcsinx=0

f(x) = 0.

Now let
us differentiate the function.

=> f'(x) =
[arcsin(3x-4x^3)]' - [3arcsinx]'

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

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

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

Therefore f'(x) = [arcsin(3x-4x^3)]' -
[3arcsinx]'

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

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

Now
as

f'(x) = 3(1 - 4x^2) sqrt (1-x^2) - 3sqrt[1 -
(3x-4x^3)^2] / sqrt[1 - (3x-4x^3)^2] *sqrt(1-x^2) = 0

It
proves that arcsin(3x-4x^3) - 3arcsinx = 0