Question
sin2 9 ° + sin2 10 ° +
sin2 11 ° + sin2 12 ° + ……… + sin2 81 ° = ?Solution
sin2 9 ° + sin2 10 ° + sin2 11 ° + sin2 12 ° + ……… + sin2 81 ° = (sin 2 9 ° + sin2 81 °)+(sin2 8 ° + sin2 82 °) + .…. + (sin2 44 ° + sin2 46 °) + sin2 45 ° = (sin2 9 ° + cos2 9 °)+(sin2 10 °+ cos2 10 °) + .…. + (sin2 44 ° + cos2 44 °)+ sin2 45 °  {As sin (90 ° - θ) = cos θ} = (1)+(1) + .…. upto 36 times + sin2 45 ° { As sin2 θ + cos2 θ= 1} = 36 + sin2 45 °                 = 36 + (1/√2)2 {As sin 45 ° =1/√2 } = 36 + 
 = 36.5
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