Question

How do you verify `[sin^3(B) + cos^3(B)] / [sin(B) + cos(B)] = 1-sin(B)cos(B)`?

Answer 1

Proof below

Explanation:

Expansion of `a^3+b^3=(a+b)(a^2-ab+b^2)`, and we can use this:
`(sin^3B+cos^3B)/(sinB+cosB)=((sinB+cosB)(sin^2B-sinBcosB+cos^2B))/(sinB+cosB)`
`=sin^2B-sinBcosB+cos^2B`
`=sin^2B+cos^2B-sinBcosB` (identity: `sin^2x+cos^2x=1`)
`=1-sinBcosB`