$\text{ The average of the numbers }n\sin n^{\circ }\text{ where }n=2,4,6,\dots {180}^{\circ }\text{ is }$

${\mathrm{S}}_{\mathrm{n}}=2\sin 2^{\circ }+4\sin 4^{\circ }+\dots +180\sin {180}^{\circ }$

$\begin{array}{l}{\mathrm{S}}_{\mathrm{n}}=2\sin 2^{\circ }+4\sin 4^{\circ }+\dots +178\sin {178}^{\circ }\\ =\left(2\sin 2^{\circ }+4\sin 4^{\circ }\stackrel{ }{+.....+}178\sin {178}^{\circ }\right)+90\\ =\left(2\sin 2^{\circ }+4\sin 4^{\circ }+\dots 176\sin 4^{°}+178\sin 2^{\circ }\right)+90\\ =180\left[\sin 2^{\circ }+\sin 4^{\circ }+\dots +\sin {88}^{\circ }\right]+90\\ =\frac{90\left[2\sin 1°\sin 2^{°}+2\sin 1^{°}\sin 4^{°}+....+2\sin 1^{°}\sin {88}^{°}\right]}{\sin 1°}+90\\ =\frac{90\left[\left(\cos 1^{\circ }-\cos 3^{\circ }\right)+\left(\cos 3^{\circ }-\cos 5^{\circ }\right)+......+\left(\cos {87}^{\circ }-\cos {89}^{\circ }\right)\right]}{\sin 1^{\circ }}+90\\ =\frac{90\left(\cos 1^{\circ }-\cos {89}^{\circ }\right)}{\sin 1^{\circ }}+90=90\left(\cot 1^{\circ }-1\right)+90\\ =90\cot 1^{\circ }-90+90\\ =90\cot 1^{\circ }\end{array}$

$\text{ Average }=\frac{90\cot 1^{\circ }}{90}=\cot 1^{\circ }$