How do you prove that #sin5x=sinx(cos^2 2x-sin^2 2x)+2cosxcos2xsin2x#?

1 Answer
sente
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(5x) = sin(x + 4x)#

#=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)#

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