Find the value of `sin^(2)5^(@)+sin^(2)10^(@)+sin^(2)15^(@)+.............+sin^(2)90^(@)`.

Explanation

`sin^(2)5^(@)+sin^(2)10^(@)sin^(2)15^(@)+.............sin^(2)90^(@)`.
`=(sin^(2)5^(@)sin^(2)85^(@))+(sin^(2)10^(@)+sin^(2)80^(@))+(sin^(2)15^(@)+sin^(2)75^(@))+(sin^(2)20^(@)+sin^(2)70^(@))+(sin^(2)25^(@)+sin^(2)65^(@))+(sin^(2)30^(@)+sin^(2)60^(@))+`
`(sin^(2)35^(@)+sin^(2)55^(@))+(sin^(2)40^(@)+sin^(2)50^(@))+sin^(2)45^(@)+sin^(2)90^(@)`
`={sin^(2)5^(@)+sin^(2)(90^(@)-15^(@))}+...................sin^(2)45^(@)+sin^(2)90^(@)`
`=(sin^(2)5^(@)+cos^(2)5^(@))+(sin^(2)10^(@)+cos^(2)15^(@))+...................+sin^(2)45^(@)+sin^(2)90^(@)`
`=1+1+1+...............` (upto 8 terms) `+((1)/(sqrt(2)))^(2)+1`
`=8+(1)/(2)+1=9(1)/(2)`.
Hence the required value `=9(1)/(2)`