sin2 7˚ + sin2 9˚ +

Solution

sin ²7˚ + sin ²9˚ + sin ²11˚ + sin ²12˚ + ……… + sin ²83˚ = (sin² 7˚ + sin² 83˚)+(sin² 9˚ + sin² 81˚) + .…. + (sin² 43˚ + sin² 47˚) + sin² 45˚ = (sin² 7˚ + cos² 7˚) + (sin² 9˚ + cos² 9˚) + .…. + (sin² 43˚ + cos²43˚) + sin² 45˚ {As sin (90˚ - θ) = cos θ} = (1)+(1) + .…. upto 19 times + sin² 45˚ { Assin ²θ +cos ²θ= 1} = 19 + sin² 45˚ {As there are 19 odd numbers from 7 to 43}. = 19 + (1/√2)² {As sin 45˚ =1/√2 } = 19 + 0.5 = 19.5 ` ` ` ` ` ` ` ` ` ` ` `