In studying the interference of light waves, the identity #sin(x)sin[(3/2)x]/sin[(1/2)x] = sin(x) + sin(2x) # is used. Prove this identity?

Question

#sin(x)sin(3/2x)/sin(1/2x )= sin(x) + sin(2x)#
Prove this identity

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Answer 1 · 2017-11-18T00:06:51

#LHS=sin(x)sin(3/2x)/sin(1/2x )#

#=sin(x)(3sin(1/2x)-4sin^3(1/2x))/sin(1/2x )#

#=sin(x)(3-2*2sin^2(1/2x))#

#=sin(x)(3-2(1-cos(x)))#

#=sin(x)(3-2+2cos(x))#

#=sin(x)(1+2cos(x))#

#=sin(x)+2sin(x)cos(x))#

#= sin(x) + sin(2x)=RHS#