(d) Given expression is

${\sin ^2}{5^o} + {\sin ^2}{10^o} + {\sin ^2}{15^o} + ….. + {\sin ^2}{85^o} + {\sin ^2}{90^o}.$

We know that $\sin {90^o} = 1$ or ${\sin ^2}{90^o} = 1$. 

Similarly, $\sin {45^o} = \frac{1}{{\sqrt 2 }}{\rm{or}}\,{\rm{si}}{{\rm{n}}^{\rm{2}}}{45^o} = \frac{1}{2}$ 

and the angles are in $A.P. $ of $18$ terms. 

We also know that ${\sin ^2}{85^o} = {[\sin ({90^o} – {5^o})]^2}$$ = {\cos ^2}{5^o}.$ 

Therefore from the complementary rule, we find 

$\therefore$ ${\sin ^2}{5^o} + {\sin ^2}{85^o} = {\sin ^2}{5^o} + {\cos ^2}{5^o} = 1.$ 

Therefore, 

${\sin ^2}{5^o} + {\sin ^2}{10^o} + {\sin ^2}{15^o} + … + {\sin ^2}{85^o} + {\sin ^2}{90^o}$ 

$ = (1 + 1 + 1 + 1 + 1 + 1 + 1 + 1) + 1 + \frac{1}{2} = 9\frac{1}{2}$.