Let x=sin(pi/7)sin((2pi)/7)sin((3pi)/7)
and
y=cos(pi/7)cos((2pi)/7)cos((3pi)/7)
So
8xy=2sin(pi/7)cos(pi/7) xx2sin((2pi)/7)cos((2pi)/7)xx2sin((3pi)/7)cos((3pi)/7)
=sin((2pi)/7)sin((4pi)/7)sin((6pi)/7)
=sin((2pi)/7)sin(pi-(3pi)/7)sin(pi-pi/7)
=sin((2pi)/7)sin((3pi)/7)sin(pi/7)=x
Hence y=1/8
Now let
cos(pi/7)=a,cos((2pi)/7)=b,cos((3pi)/7)=c
So
y=cos(pi/7)cos((2pi)/7)cos((3pi)/7)=abc=1/8
Again
8x^2=8sin^2(pi/7)sin^2((2pi)/7)sin^2((3pi)/7)
=[1-cos((2pi)/7)][1-cos((4pi)/7)][1-cos((6pi)/7)]
=[1-cos((2pi)/7)][1+cos((3pi)/7)][1+cos(pi/7)]
=(1-b)(1+c)(1+a)
=(1-b)(1+c)(1+a)
=1+a-b+c+ac-ab-bc-abc
=1+a-b+c+ac-ab-bc-1/8
=>8x^2=7/8+a-b+c+ac-ab-bc
Now
ac-ab-bc=1/2(2ac-2ab-2bc)
=1/2[2cos(pi/7)cos((3pi)/7)
-2cos(pi/7)cos((2pi)/7)-2cos((2pi)/7)cos((3pi)/7)]
=1/2[cos((4pi)/7)+cos((2pi)/7)
-cos((3pi)/7)-cos(pi/7)-cos((5pi)/7)-cos(pi/7)]
=1/2[-cos((3pi)/7)+cos((2pi)/7)
-cos((3pi)/7)+cos((2pi)/7)-2cos(pi/7)]
=1/2[2cos((2pi)/7)
-2cos((3pi)/7)-2cos(pi/7)]
=b-a-c
=>ac-ab-bc+a-b+c=0
Hence we get
=>8x^2=7/8+a-b+c+ac-ab-bc
=>8x^2=7/8+0
=>x^2=7/64
=>x=sqrt7/8
=>sin(pi/7)sin((2pi)/7)sin((3pi)/7)=sqrt7/8
