Question

How do you prove `sin^3 x - cos^3 x = (1 + sin x cos x )( sin x cos x )`?

Answer 1

This is a mistaken identity. Substitute `x=0` - left part equals to `cos^3(0)=-1` while the right part equals to `0` since `sin(0)=0`.
The right identity is
`sin^3x-cos^3x=(1+sinx*cosx)(sinx-cosx)`

Explanation:

Recall the trivial algebraic formula
`a^3-b^3=(a-b)(a^2+ab+b^2)`
It can be verified by direct multiplication in the right part.

Applying this to our problem for `a=sinx` and `b=cosx`, we obtain
`sin^3x-cos^3x=(sinx-cosx)(sin^2x+sinx*cosx+cos^2x)`

Since `sin^2x+cos^2x=1`, the expression in the second pair of parenthesis in the right part of this identity can be simplified:
`sin^2x+sinx*cosx+cos^2x=1+sinx*cosx`.
This completes the proof.