proving trigonometri equation?

Question

prove
cos 3t + sin 3t = (cos t - sin t)(1+2 sin 2t)

Answer 1

LHS = cos 3t + sin 3t ( make 3t = 2t+t as 3t is not on RHS
= cos(2t+t) + sin (2t+t)
= cos 2t cos t - sin 2t sin t + sin 2t cos t + cos 2t sin t
= cos2t(cos t + sin t) + sin 2t(cos t - sin t)
do not distrub the 2nd term as it is in rHS
cos 2t(cos t+sin t) = (cos^2 t- sin ^2 t)(cos t+ sin t)
= (cos t- sin t)(cost+sin t)( cost + sin t)
= (cost - sint)(cos^2 t + sin ^2 t + 2 sin t cos t)
= (cos t- sin t) (1+ sin 2t)
adding 2nd term we get

LHS = (cos t - sin t)(1+ 2 sin 2t) = RHS

Answer 2

(cos t - sin t)(1 + 2 sin 2t)
= cos t - sin t + 2 cos t sin 2t - 2 sin t sin 2t
= cos t - sin t + cos t sin 2t - sin t sin 2t
+ 2 cos^2 t sin t - 2 sin^2 t cos t
= sin t(2 cos^2 t - 1) + cos t(1 - 2 sin^2 t) + cos t sin 2t - sin t sin 2t
= sin t cos 2t + cos t cos 2t + cos t sin 2t - sin t sin 2t
= [cos 2t cos t - sin 2t sin t] + [sin t cos 2t + cos t sin 2t]
= cos 3t + sin 3t