Subtracting the identities sin( x + y ) = sin(x) cos (y) + cos(x) sin(y),
and sin(x-y) = sin(x) cos(y) - cos(x) sin(y) gives us the identity:
2cos(x)sin(y) = sin(x+y) - sin(x-y).
So, we have:
2cos(p/7)sin(p/7)=sin(2p/7)-sin(0)
2cos(3p/7)sin(p/7)=sin(4p/7)-sin(2p/7)
2cos(5p/7)sin(p/7)=sin(6p/7)-sin(4p/7)
Adding these three equations gives:
2sin(p/7)[cos(p/7) + cos(3p/7) + cos(5p/7)] = sin(6p/7),
which gives the result. |
