Question #c71e9

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Answer 1 · 2017-12-18T06:43:27

#cos 22.5 = sqrt ( (sqrt2 + 2 )/4) #

For this problem we must consider one of our trigonometric identities:

#cos 2theta -= cos ^2 theta -sin^2 theta #

We can then use our identity:

#cos^2 theta + sin^2 theta -= 1 #

#=> cos 2 theta = 2cos^ 2theta -1 #

Letting #theta = 22.5 #:

#=> cos 45 = 2cos^2 22.5 -1 #

We know #cos45= sqrt2 / 2 #

#=>sqrt2 /2 = 2cos^2 22.5 -1 #

#=>sqrt2/2 +1 = 2 cos^2 22.5 #

#=> (sqrt2 + 2)/4 = cos^2 22.5 #

#=> sqrt ( (sqrt2 + 2 )/4) = cos22.5 #