Show that Cos 22.5=√2+√2/2

Answer 1

22.5° = 45° / 2

cos 45° = √ 2 / 2

cos ( θ / 2 ) = ± √ [ ( 1 + cos θ ) / 2 ]

45° is located in first quadrant where all trigonometric functions are positive so:

cos 22.5° = cos ( 45° / 2 ) = √ [ ( 1 + cos 45° ) / 2 ] =

√ [ ( 1 + √ 2 / 2 ) / 2 ] = √ [ ( 2 / 2 + √ 2 / 2 ) / 2 ] =

√ [ ( 2 + √ 2 ) / 2 / 2 ] = √ [ ( 2 + √ 2 ) / 4 ] =

√ ( 2 + √ 2 ) / √ 4 = √ ( 2 + √ 2 ) / 2