How do you use the half angle identity to find exact value of cos^2(pi/12)?

1 Answer
Nghi N · Nghi N.
Aug 17, 2015

Find #cos^2 (pi/12)#

Ans: #(2 + sqrt3)/4#

Explanation:

Call #cos (pi/12) = cos t#
#cos 2t = cos ((2pi)/12) = cos (pi/6) = sqrt3/2#
a. Apply the double angle identity:
#cos 2t = 2cos^2 t - 1#
#2cos^2 t = 1 + cos 2t = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 t = cos^2 (pi/12) = (2 + sqrt3)/4#
b. Apply the half angle identity:
#cos^2 (t/2) = (1 + cos t)/2#
In this case cos t = cos (pi/6) = sqrt3/2
#cos^2 (pi/12) = (1 + sqrt3/2)/2 = (2 + sqrt3)/4#

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