Quote by crash_matrix
sum[0 to x](sin(x))=180 * sin(x/2)*sin(x/2) + sin(x)/2
where x is in degrees

Didn't you mean f(180), instead of 180, above? f(180) being the sum[t=0 to 180](sin(t)).
Also, to avoid confusing the two meanings of 'x', you may want to express the lefthand side as sum[t=0 to x](sin(t)) ; 'x' is the limit of the sum, not really the argument to sin().
The x's in the righthand side are correct, as they mean the same value of 'x' as the limit of the sum in the lefthand side.
So, if I get you right, the conjecture is[tex]\sum_{t=0}^x \sin t = \left( \sum_{t=0}^{180} \sin t \right) \sin^2(x/2) + (\sin x)/2[/tex]with the arguments in degrees, not radians.
The two overlapping curves (lefthand side, righthand side) looking like this, for x=0 to 180: