How do you use a half-angle formula to simplify #tan ((7pi)/8)#?

1 Answer
Nghi N.
Jun 30, 2015

Simplify #tan ((7pi)/8)#

Explanation:

Call #tan ((7pi)/8) = t#
Use trig identity: #tan 2a = (2tan a)/(1 - tan^2 a)#

#tan ((14pi)/8) = (2t)/(1 - t^2)#

Since #tan ((14pi)/8) = tan ((6pi)/8 + pi) = tan ((3pi)/4 + pi) #=

#= - tan ((3pi)/4) = -1#

Therefor, #(2t)/(1 - t^2) = -1 # -> #t^2 - 2t - 1 = 0#
Solve this quadratic equation.
#D = d^2 = b^2 - 4ac = 4 + 4 = 8 -> d = +- 2sqrt2#
#t = 1 +- sqrt2#

#t = tan ((7pi)/8) = 1 - sqrt2# (Quadrant II)

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