Given:
cos(alpha) cos(beta-alpha) - sin(alpha) sin(beta-alpha) = cos(beta)
Substitute cos(beta-alpha) = cos(beta)cos(alpha)+sin(beta)sin(alpha):
cos(alpha)(cos(beta)cos(alpha)+sin(beta)sin(alpha)) - sin(alpha) sin(beta-alpha) = cos(beta)
Distribute cos(alpha) across the terms within the parentheses:
cos(beta)cos^2(alpha)+sin(beta)sin(alpha)cos(alpha) - sin(alpha) sin(beta-alpha) = cos(beta)
Substitute sin(beta-alpha) = sin(beta)cos(alpha)-cos(beta)sin(alpha):
cos(beta)cos^2(alpha)+sin(beta)sin(alpha)cos(alpha) - sin(alpha)(sin(beta)cos(alpha)-cos(beta)sin(alpha)) = cos(beta)
Distribute -sin(alpha) across the terms within the parentheses:
cos(beta)cos^2(alpha)+sin(beta)sin(alpha)cos(alpha) - sin(beta)sin(alpha)cos(alpha)+cos(beta)sin^2(alpha) = cos(beta)
Please observe that the two terms in color(red)("red") sum to 0:
cos(beta)cos^2(alpha)+color(red)(sin(beta)sin(alpha)cos(alpha) - sin(beta)sin(alpha)cos(alpha))+cos(beta)sin^2(alpha) = cos(beta)
Rewriting without the two terms:
cos(beta)cos^2(alpha)+cos(beta)sin^2(alpha) = cos(beta)
Factor out cos(beta):
cos(beta)(cos^2(alpha)+sin^2(alpha)) = cos(beta)
We know that (cos^2(alpha)+sin^2(alpha)) = 1, therefore, we need not write the factor:
cos(beta) = cos(beta) Q.E.D.