Prove:
sin^3 X+cos^4 X=1

Here is what I have done so far:
Prrof:
sin^3x+cos^4x=1
sin^3x=1-cos^4x
sin^3x=(1-cos^2x)(1+cos^2x)
sin^3x=sin^2x(1+cos^2x)
sin^3x-[sin^2x(1+cos^2x)]=0
sin^2x[sinx-1(1+cos^2x)]=0

I do not know where to go from here or if I am on the right track... help.

Are you sure that the question is:

Prove:
sin^3 X+cos^4 X=1?

It is clear to me that you have read the question incorrectly.

May be the question is:

SOLVE: sin^3 X+cos^4 X=1

The above equation is not correct for all values of X.

The only way above equation can be true is if

Sin x = 1 or sin x = 0

Which is possible, e.g. if x = pi/2 or x = 0