sin²5° + sin²6° + ............. sin²84° + sin²85° = ?

Solution(By Examveda Team)

$$\eqalign{ & {\sin ^2}{5^ \circ } + {\sin ^2}{6^ \circ } + \,.....\,{\sin ^2}{84^ \circ } + {\sin ^2}{85^ \circ } \cr & {\text{Number of terms}} \cr & = \frac{{85 - 5}}{1} + 1 \cr & = 80 + 1 \cr & = 81 \cr} $$
  $$ = \left( {{{\sin }^2}{5^ \circ } + {{\sin }^2}{{85}^ \circ }} \right)\, + $$     $$\,\left( {{{\sin }^2}{6^ \circ } + {{\sin }^2}{{84}^ \circ }} \right)\, + $$     . . . . . upto 40 pairs $$ + $$ middle term
$$\eqalign{ & = {\text{40}} + {\sin ^2}{45^ \circ } \cr & = 40 + \frac{1}{2} \cr & = 40\frac{1}{2} \cr} $$

Alternate:
In case,
when series is in the form of sin²θ or cos²θ,
then sum of series will always be half of number of terms.
$$ = \frac{{81}}{2} = 40\frac{1}{2}$$