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

1 Answer
Jun 18, 2015

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.