sin(A + B)

sin A = cos(pi/2 - A)

sin (A + B) = cos(pi/2 - (A + B))

sin(A + B) = cos ((pi/2 - A) - B)

sin(A + B) = cos(pi/2 - A)cosB + sin(pi/2 - A)sinB

sin(A + B) = sinAcosB + cosAsinB

sin 2A

sin 2A = sin(A + A)

sin 2A = sinAcosA + cosAsinA

sin 2A = 2sinAcosA

sin(A - B)

sin(A - B) = sin(A + -B)

sin(A - B) = sinAcos(-B) + cosAsin(-B)

sin(A - B) = sinAcosB - cosAsinB