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