How do you prove that #sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x#?
1 Answer
Oct 22, 2016
Using the angle addition identity:
#sin(alpha + beta) = sin(alpha)cos(beta) + cos(alpha)sin(beta)#
along with the double angle identities:
#sin(2theta) = 2sin(theta)cos(theta)# #cos(2theta) = cos^2(theta)-sin^2(theta)#
we have
#=sin(x)cos(4x) + cos(x)sin(4x)#
#=sin(x)cos(2*2x) + cos(x)sin(2*2x)#
#=sin(x)(cos^2(2x)-sin^2(2x)) + cos(x)(2sin(2x)cos(2x))#
#=sin(x)(cos^2(2x)-sin^2(2x))+2cos(2x)cos(2x)sin(2x)#