How do you find the exact values of cos 22.5 degrees using the half angle formula?

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Aug 6, 2015

The half angle identity for cosine can be derived (since I don't recall it off-hand):

#cos^2(x) = (1+cos(2x))/2#

By inference:
#cos^2(x/2) = (1+cosx)/2#

Square root to get:

#cos(x/2) = pmsqrt((1+cosx)/2)#
#+# if quadrant I or IV
#-# if quadrant II or III

#22.5^o# is quadrant I, so it is positive.

#cos(45^o/2) = sqrt((1+cos45^o)/2)#

#= sqrt((1+(sqrt2/2))/2)#

#= sqrt((((2+sqrt2)/2))/2)#

#= sqrt((2+sqrt2)/4)#

#= color(blue)(sqrt(2+sqrt2)/2)#

or #~~0.9238795#

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