Question

What is the exact value of cos 22.5°?

Answer 1

`cos( (x)/(2) )= \pm \sqrt{ (1+cosx)/(2) } \\\\ cos( (45)/(2) )= \pm \sqrt{ (1+cos(45))/(2) } \\\\ cos( 22.5)= \pm \sqrt{ (1+ ( √(2) )/(2) )/(2) } \\\\ cos( 22.5)= \pm \sqrt{ ( (2+ √(2) )/(2) )/(2) } \\\\ cos( 22.5)= \pm \sqrt{ {( 2+√(2) )/(2)* (1)/(2) }} \\\\ cos( 22.5)= \pm \sqrt{ {( 2+√(2) )/(4)}} \\\\ cos(22.5)= \pm \frac{ \sqrt{2+ √(2) } }{2}`