The value of sin²2° + sin²4° + sin²6° + ........ + sin²90° is?

Solution (By Examveda Team)

$$\eqalign{ & {\text{According to the question,}} \cr & {\sin ^2}{2^ \circ } + {\sin ^2}{4^ \circ } + {\sin ^2}{6^ \circ } + ..... + {\sin ^2}{90^ \circ } \cr & {\text{Number of terms}} \cr & = \frac{{l - a}}{d} + 1 \cr & = \frac{{90 - 2}}{2} + 1 \cr & = 45 \cr & {\text{But }}{\sin ^2}{90^ \circ } = 1 \cr & {\text{So, 22 pairs}} + {\sin ^2}{90^ \circ } \cr & = 22 + 1 \cr & = 23 \cr} $$