The average of the numbers $n\sin n°$ for $n=2,4,6,\cdots ,180$

Let $S=2\sin 2°+4\sin 4°+\cdots +178\sin 178°+180\sin 180°$

$S=2[\sin 2°+2\sin 4°+\cdots +89\sin 178°]\to \left(i\right)$

$S=2[89\sin 178°+88\sin 176°+\cdots +1.\sin 2°]\to \left(ii\right)$

(writing in reverse order)

$\left(i\right)+\left(ii\right)$

$\Rightarrow 2S=2\times 90[\sin 2°+\sin 4°+\cdots +\sin 178°]$

$S=90\cdot \frac{\sin 89°}{\sin 1°}\cdot \sin 90°$

$S=90\cot 1°$

Average value $=\frac{90\cot 1°}{90}=\cot 1°$.