Explanation

(c)
$\sin^{2}{5{^\circ} + \sin^{2}{10{^\circ} + \sin^{2}{15{^\circ} + ... + \sin^{2}{90{^\circ}}}}}$
$= \sin^{2}{5{^\circ} + \sin^{2}{10{^\circ} + {\ldots + \sin^{2}}{45{^\circ} + ... + \sin^{2}{80{^\circ} + \sin^{2}{85{^\circ}}} + \sin^{2}{90{^\circ}}}}}$
$= \sin^{2}{5{^\circ} + \sin^{2}{10{^\circ} + ... + \frac{1}{2} + {\ldots + \cos^{2}}{10{^\circ}\mathbf{+}cos^{2}5{^\circ} + \sin^{2}{90{^\circ}}}}}$
$= \left( \sin^{2}{5{^\circ} + \cos^{2}{5{^\circ}}} \right) + (\sin^{2}{10{^\circ} + \cos^{2}{10{^\circ}}) + \ldots} +$
$\left( \sin^{2}{40{^\circ} + \cos^{2}{40{^\circ}}} \right) + \sin^{2}{45{^\circ}} + \sin^{2}{90{^\circ}}$
$= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \frac{1}{2} + 1 = 9\frac{1}{2}$