cos 11π 8
Solution
Let α = 11π 8
cos 2α = 2 cos2 α - 1     (Double angle formula)
cos 2( 11π 8 ) = 2 cos2 ( 11π 8 ) - 1
cos ( 11π 4 ) = 2 cos2 ( 11π 8 ) - 1
1 + cos ( 11π 4 ) = 2 cos2 ( 11π 8 )
2 cos2 ( 11π 8 ) = 1 + cos ( 11π 4 )
= 1 + cos ( (8 + 3)π 4 )
= 1 + cos ( 4 + 4 )
= 1 + cos (2π + 4 )
= 1 + (cos 2π . cos 4 - sin 2π . sin 4 )
2 cos2 ( 11π 8 ) = 1 + (1 . -1 2   -   0 . 1 2 )
2 cos2 ( 11π 8 ) = 1 - 1 2
= 2   - 1 2
= 2 - √ 2 2
cos2 ( 11π 8 ) = 2 - √ 2 4
± 2 - 2 2 = cos 11π 8
      ∴ cos 11π 8      =       ± 2 - √ 2 2

sin 11π 8
Solution
Let α = 11π 8
cos 2α = 1 - 2 sin2 α     (Double angle formula)
cos 2( 11π 8 ) = 1 - 2 sin2 ( 11π 8 )
cos ( 11π 4 ) = 1 - 2 sin2 ( 11π 8 )
2 sin2 ( 11π 8 ) + cos ( 11π 4 ) = 1
2 sin2 ( 11π 8 ) = 1 - cos ( 11π 4 )
= 1 - cos ( (8 + 3)π 4 )
= 1 - cos ( 4 + 4 )
= 1 - cos ( 2π + 4 )
= 1 - ( cos 2π . cos 4  -   sin 2π . sin 4 )
2 sin2 ( 11π 8 ) = 1 - (1 . -1 2   -   0 . 1 2 )
2 sin2 ( 11π 8 ) = 1 - -1 2
= 2   + 1 2
= 2 + √ 2 2
sin2 ( 11π 8 ) = 2 + √ 2 4
± 2 + 2 2 = sin 11π 8
      ∴ sin 11π 8      =       ± 2 + √ 2 2