How do you find the exact values of tan 67.5 degrees using the half angle formula?

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Answer 1 · 2015-07-30T04:14:40

Find tan (67.5) deg

Ans: #1 + sqrt2#

Call tan (67.5) = t .

#tan 2t = tan (135) = tan (-45 + 180) = - tan (45) = - 1#

Use trig identity: #tan 2t = -1 = (2t)/(1 - t^2)#

t^2 - 2t - 1 = 0

#D = d^2 = b^2 - 4ac = 4 + 4 = 8 #--> #d +- 2sqrt2#

#t = tan (67.5) = 1 +- sqrt2#
Since the arc (67.5) is in Quadrant I, only the positive number is accepted -> #tan (67.5) = 1 + sqrt2#