cos ^-π⁄ ₈
Solution
Let α = ^-π⁄ ₈
cos 2α = 2 cos² α - 1     (Double angle formula)

cos 2( -π⁄ 8 )	=	2 cos2 ( -π⁄ 8 ) - 1
cos ( -π⁄ 4 )	=	2 cos2 ( -π⁄ 8 ) - 1
1 + cos ( -π⁄ 4 )	=	2 cos2 ( -π⁄ 8 )
1 + 1⁄ √ 2	=	2 cos2 ( -π⁄ 8 )
√ 2 + 1 ⁄ √ 2	=	2 cos2 ( -π⁄ 8 )
2 + √ 2 ⁄2	=	2 cos2 ( -π⁄ 8 )
2 + √ 2 ⁄4	=	cos2 ( -π⁄ 8 )

cos ( ^-π⁄ ₈ ) = ± ^{√ 2 + √ 2} ⁄₂

sin ^-π⁄ ₈
Solution
Let α = ^-π⁄ ₈
cos 2α = 1 - 2 sin² α     (Double angle formula)

cos 2( -π⁄ 8 )	=	1 - 2 sin2 ( -π⁄ 8 )
cos ( -π⁄ 4 )	=	1 - 2 sin2 ( -π⁄ 8 )
2 sin2 ( -π⁄ 8 ) + cos ( -π⁄ 4 )	=	1
2 sin2 ( -π⁄ 8 )	=	1 - cos ( -π⁄ 4 )
	=	1 - ( -1⁄ √ 2 )
	=	1 - 1⁄ √ 2
	=	√ 2   - 1 ⁄ √ 2
	=	2 - √ 2 ⁄2
sin2 ( -π⁄ 8 )	=	2 - √ 2 ⁄ 4

      ∴ sin ^-π⁄ ₈      =       ± ^{√ 2 - √ 2} ⁄₂