Mean of the values sin 2 10 ° , sin 2 20 ° , sin 2 30 ° , … … … , sin 2 90…

We know that, $\mathrm{Mean}=\frac{\mathrm{Total} \mathrm{Sum}}{\mathrm{Number} \mathrm{of} \mathrm{terms}}$

Total sum $={\sin }^{2}10°+{\sin }^{2}20°+{\sin }^{2}30°+....+{\sin }^{2}90°$

Using $\sin x=\cos \left(90-x\right)$, we get

$={\sin }^{2}10°+{\sin }^{2}20°+{\sin }^{2}30°+....+{\sin }^2\left(90°-20°\right)+{\sin }^2\left(90°-10°\right)+{\sin }^{2}90°$

$={\sin }^{2}10°+{\sin }^{2}20°+{\sin }^{2}30°+....+{\cos }^{2}20°+{\cos }^{2}10°+{\sin }^{2}90°$

$=1+1+1+1+1$ $\left(\because {\sin }^{2}x+{\cos }^{2}x=1, \sin 90°=1\right)$

$=5$

Total number of terms $=9$

Hence, Mean $=\frac{5}{9}$