sin3A
=sin(2A+A)
=sin2AcosA+cos2AsinA
=(2sinAcosA)cosA+sinA(cos^2 A-sin^2 A)
=2sinAcos^2 A+sinAcos^2 A-sin^3 A
=3cos^2 AsinA-sin^3 A
=3(1-sin^2 A)sinA-sin^3 A ***
=3sinA-3sin^3 A-sin^3 A
=3sinA-4sin^3 A
since LHS=RHS
this trigonometry is proven
*** since cos^2 A+sin^2 A=1
thus cos^2 A=1-sin^2 A