Question

How do you verify `(sin^3x+cos^3x)/(sinx+cosx)=1-sinxcosx`?

Answer 1

Proof below

Explanation:

Expansion of a cubic `a^3+b^3=(a+b)(a^2-ab+b^2)`
`(sin^3x+cos^3x)/(sinx+cosx)=((sinx+cosx)(sin^2x-sinxcosx+cos^2x))/(sinx+cosx)`
`=sin^2x-sinxcosx+cos^2x`
Identity: `sin^2x+cos^2x=1`
`=sin^2x+cos^2x-sinxcosx`
`=1-sinxcosx`