How do you find the value of #sec((5pi)/12)# using the double angle or half angle identity?

1 Answer
Nghi N.
Jun 24, 2016

#2/(sqrt(2 + sqrt3)#

Explanation:

#sec ((5pi)/12) = 1/(cos((5pi)/12) #.
Find #cos ((5pi)/12)# by using trig identity:
#cos 2a = 2cos^2 a - 1#
Call #cos ((5pi)/12) = cos t#, we get:
#cos ((10pi)/12) = cos (pi/6) = sqrt3/2 = 2cos^2 t - 1#
#2cos^2 t = 1 + sqrt3/2 = (2 + sqrt3)/2#
#cos^2 t = (2 + sqrt3)/4#
#cos ((5pi)/12) = cos t = sqrt(2 + sqrt3)/2.#
Since cos ((5pi)/12) is positive, therefor, the positive value is accepted. Finally,
#sec ((5pi)/12) = 1/(cos) = 2/(sqrt(2 + sqrt3)#

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