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Question

sin2(αβ)+sin2(βγ)+sin2(γα) =4sin(αβ)sin(βγ)sin(γα).

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Solution

Let αβ=x,βγ=y,γα=z.
x+y+z=0 or x+y=z
sin(x+y)=sinz,cos(c+y)=cosz .(1)
Hence we have to prove that
sin2x+sin2y+sin2z=4sinxsinysinz.
L.H.S. =2sin(x+y)cos(xy)+2sinzcosz by (1)
=2sinzcos(xy)+2sinzcosz
=2sinz[cos(xy)cosz]
=2sinz[cos(xy)cos(x+y)], by (1)
=2sinz(2sinxsiny).
=4sinxsinysinz.
=4sin(αβ)sin(βγ)sin(γα).

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