S＝sinx+sin2x+sin3x+…+sinnx　とおいて
両辺に　2sin(\(\frac{x}{2}\))　をかけると
2sin(\(\frac{x}{2}\))S=2sin(\(\frac{x}{2}\))sinx+2sin(\(\frac{x}{2}\))sin2x+2sin(\(\frac{x}{2}\))sin3x+…+2sin(\(\frac{x}{2}\))sinnx

2sinαsinβ＝cos(α-β)-cos(α+β)　を利用して

2sin(\(\frac{x}{2}\))S＝[cos(\(\frac{x}{2}\))- cos(3\(\frac{x}{2}\))]+[cos(3\(\frac{x}{2}\))-cos5\(\frac{x}{2}\))]
　　　+[cos(5\(\frac{x}{2}\))-cos(7\(\frac{x}{2}\))]+…+{cos[(2n-1)\(\frac{x}{2}\)]- cos[(2n+1)\(\frac{x}{2}\)]}

2sin(\(\frac{x}{2}\))S=cos(\(\frac{x}{2}\))-cos[(2n+1)\(\frac{x}{2}\)］

あとは両辺を2sin(\(\frac{x}{2}\))で割って終わりです。