$\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}90^{\circ}$
$\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}85^{\circ}+1$
Number of terms in $5,10,15,20,\dots 85$
$\implies l = a + (n-1)d,\:$Where $l = $ last terms, $a = $ first term, $n = $ total number of terms
$\implies 85 = 5 + (n-1)5$
$\implies n = 17$
We know that $\sin^{2}\theta + \:cos^{2}\theta = 1$
$\Large \star$ Every pairs gives value $'1'.$
$\:\:[\because \sin^{2} 5^{\circ} + \sin^{2} 85^{\circ} = \sin^{2} 5 ^{\circ} + \sin^{2} (90^{\circ}-5^{\circ}) = \sin^{2} 5 ^{\circ} + \cos^{2} 5 ^{\circ} = 1].$
$\therefore\text{Number of pairs} = \dfrac{\text{Number of terms}}{2} = \dfrac{n}{2} = \dfrac{17}{2} = 8.5$
Now, $\underbrace{\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}85^{\circ}}_{8.5}+1 = 8.5 + 1 = 9.5$
So, the correct answer is $(C).$