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

1 Answer

May 22, 2015

Use the trig identity: #tan^2 x = (1 - cos 2x)/(1 + cos 2x)#

#tan^2 ((7pi)/8) = (1 - cos (7pi)/4)/(1 + cos (7pi)/4# =

or #cos ((7pi)/4)= cos (pi/4) = 1/sqrt2#, then:

#tan^2 (7pi/8) = (sqrt2 - 1)/(sqrt2+ 1)#=

#= (sqrt2 - 1)^2/(2 - 1) = (sqrt2 - 1)^2#

Finally,# tan ((7pi)/8) = (sqrt2 - 1)#

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