So many sins in this sineful problem !!!!!

Geometry Level pending

If a , b , c ( 0 , π ) a,b,c∈(0,π) and a + b + c = π a+b+c=π ,

Then, Find the value of

sin ( b ) . sin ( c ) . sin ( b c ) ( sin 2 ( b ) + sin 2 ( c ) + sin 2 ( b c ) ) + sin ( c ) . sin ( a ) sin ( c a ) ( sin 2 ( c ) + sin 2 ( a ) + sin 2 ( c a ) ) + sin ( a ) . sin ( b ) . sin ( a b ) ( sin 2 ( a ) + sin 2 ( b ) + sin 2 ( a b ) ) + sin ( b c ) sin ( c a ) sin ( a b ) ( sin 2 ( b c ) + sin 2 ( c a ) + sin 2 ( a b ) ) \begin{aligned} &\sin(b).\sin(c).\sin(b-c)(\sin^2(b)+\sin^2(c)+\sin^2(b-c))+ \\ &\sin(c).\sin(a)\sin(c-a)(\sin^2(c)+\sin^2(a)+\sin^2(c-a))+ \\ &\sin(a).\sin(b).\sin(a-b)(\sin^2(a)+\sin^2(b)+ \sin^2(a-b))+ \\ &\sin(b-c)\sin(c-a)\sin(a-b)(\sin^2(b-c)+\sin^2(c-a)+\sin^2(a-b)) \end{aligned}


The answer is 0.

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1 solution

Cody Johnson
May 25, 2014

This problem inspired my new note that I'll be posting shortly: guessing.

Yeah.... guessing.... this problem is a worth 10 seconds one right ?

Aditya Raut - 7 years ago

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