Докажи тождества 1) {\sin \alpha + \sin 2\alpha + \sin 3\alpha + \ldots + \sin n\alpha \,\mathrlap{\,=}} {= \dfrac{\sin \dfrac{ n\alpha}{2}\sin \dfrac{ (n+1)\alpha}{2}}{\sin\dfrac{\alpha}{2}}}; 2) {\cos \alpha + \cos 2\alpha + \cos 3\alpha + \ldots + \cos n\alpha \,\mathrlap{\,=}} {=\dfrac{\sin \dfrac{n\alpha}{2}\cos \dfrac{(n+1)\alpha}{2}}{\sin\dfrac{\alpha}{2}}}.

Задание

Докажи тождества

\({\sin \alpha + \sin 2\alpha + \sin 3\alpha + \ldots + \sin n\alpha \,\mathrlap{\,=}}\) \({= \dfrac{\sin \dfrac{ n\alpha}{2}\sin \dfrac{ (n+1)\alpha}{2}}{\sin\dfrac{\alpha}{2}}}\) ;
\({\cos \alpha + \cos 2\alpha + \cos 3\alpha + \ldots + \cos n\alpha \,\mathrlap{\,=}}\) \({=\dfrac{\sin \dfrac{n\alpha}{2}\cos \dfrac{(n+1)\alpha}{2}}{\sin\dfrac{\alpha}{2}}}\) .