Question

prove without using trigonometric table
sin² 5°+ sin² 10°+....+ sin² 85°+ sin² 90° = 9 \frac{1}{2}

Answer 1

$sin^2\ 5\ +\ sin^2\ 85\ =\ sin^2\ 5 + cos^2\ (90-85)\ =\ sin^2\ 5 + cos^2\ 5\ =\ 1$

so we have pairs: 5,85; 10,80, .... 40,50;  and only one sin² 45 and one sin² 90

So the

$Sum = 8 * 1 + (\frac{1}{\sqrt{2}})^2 + 1^2 = 9 \frac{1}{2}$