cos ^7π⁄ ₈
Solution
Let α = ^7π⁄ ₈
cos 2α = 2 cos² α - 1     (Double angle formula)

cos 2( 7π⁄ 8 )	=	2 cos2 ( 7π⁄ 8 ) - 1
cos ( 7π⁄ 4 )	=	2 cos2 ( 7π⁄ 8 ) - 1
1 + cos ( 7π⁄ 4 )	=	2 cos2 ( 7π⁄ 8 )
2 cos2 ( 7π⁄ 8 )	=	1 + cos ( 7π⁄ 4 )
	=	1 + cos ( (3 + 4)π⁄ 4 )
	=	1 + cos ( 3π⁄ 4 + 4π⁄4 )
	=	1 + cos ( 3π⁄ 4 + π )
	=	1 + ( cos 3π⁄ 4.cos π - sin 3π⁄4.sin π )
2 cos2 ( 7π⁄ 8 )	=	1 + ( -1⁄ √ 2 .(-1) - 1⁄ √ 2 .0 )
2 cos2 ( 7π⁄ 8 )	=	1 + 1⁄ √ 2
	=	√ 2   + 1⁄ √ 2
	=	2 + √ 2 ⁄ 2
cos2 ( 7π⁄ 8 )	=	2 + √ 2 ⁄ 4

      ∴ cos ^7π⁄ ₈      =       ± ^{√ 2 + √ 2} ⁄₂

sin ^7π⁄ ₈
Solution
Let α = ^7π⁄ ₈
cos 2α = 1 - 2 sin² α     (Double angle formula)

cos 2( 7π⁄ 8 )	=	1 - 2 sin2 ( 7π⁄ 8 )
cos ( 7π⁄ 4 )	=	1 - 2 sin2 ( 7π⁄ 8 )
2 sin2 ( 7π⁄ 8 ) + cos ( 7π⁄ 4 )	=	1
2 sin2 ( 7π⁄ 8 )	=	1 - cos ( 7π⁄ 4 )
	=	1 - cos ( (3 + 4)π⁄ 4 )
	=	1 - cos ( 3π⁄ 4 + 4π⁄4 )
	=	1 - cos ( 3π⁄ 4 + π )
	=	1 - ( cos 3π⁄ 4.cos π - sin 3π⁄4.sin π )
2 sin2 ( 7π⁄ 8 )	=	1 - ( -1⁄ √ 2 .(-1) - 1⁄ √ 2 .0 )
2 sin2 ( 7π⁄ 8 )	=	1 - 1⁄ √ 2
	=	√ 2   - 1⁄ √ 2
	=	2 - √ 2 ⁄ 2
sin2 ( 7π⁄ 8 )	=	2 - √ 2 ⁄ 4

      ∴ sin ^7π⁄ ₈      =       ± ^{√ 2 - √ 2} ⁄₂