Question

What is the exact value of cos 22.5°?

Answer 1

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