We'll multiply both sides by sin (
/7)
sin( /7)cos(2
/7)+sin(
/7)cos(4
/7)+sin( /7)cos(6
/7) = -sin(
/7)/2
We'll
transform the product in sum:
(1/2)[sin( /7-2
/7) +
sin( /7+2
/7)] + (1/2)[sin(
/7-4
/7) + sin(
/7+4
/7)] +
(1/2)[sin( /7-6
/7) + sin(
/7+6
/7)]= -sin(
/7)/2
We'll divide by (1/2) both
sides:
-sin( /7) + sin(3
/7) - sin(3
/7) +
sin(5 /7) - sin(5
/7) + sin(7
/7) = -sin(
/7)
We'll eliminate like
terms:
-sin( /7) + sin(
)= -sin(
/7)
But sin( ) =
0
-sin( /7) = -sin(
/7)
Since we've get the same results both
sides, the given identity is verified.
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