Question

sin (alpha-beta) cos (alpha + beta) = sin alpha cos alpha - sin beta cos beta verify that each equation is an identity

Answer 1

sin(x - y) = sin(x)cos(y) - cos(x)sin(y)
cos(x + y) = cos(x)cos(y) - sin(x)sin(y)
Multiply sin(x - y) and cos(x + y)
sin(x - y)cos(x + y) = (sin(x)cos(y) - cos(x)sin(y))*(cos(x)cos(y) - sin(x)sin(y))
= sin(x)cos^2(y)cos(x) - cos^2(x)sin(y)cos(y) -sin^2(x)cos(y)sin(y) +sin(x)cos(x)sin^2(y)
= sin(x)cos(x)(sin^2(y) +cos^2(y)) - sin(y)cos(y)(sin^2(x) +cos^2(x))
= sin(x)cos(x)- sin(y)cos(y) (sin^2(a) +cos^2 (a) = 1)