Double Angle Formula Proving?

Question

The question is:
Let #t=tan(135/2)#, then use the double angle formula for #tan2theta# to show that #t^2-2t-1=0#

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Answer 1 · 2018-05-28T12:54:37

#tan 2 theta = {2 tan theta}/{1 - tan^2 theta }#

#t = tan (135^circ/2)#

#tan 135^circ = {2t}/{1-t^2}#

#-1 = {2t}/{1-t^2}#

#-1 + t^2 = 2t #

#t^2 - 2t - 1 = 0 quad sqrt#

We can go on and write

# t= 1 pm sqrt{2}#

#135^circ/2# is in the first quadrant so we can choose the positive #t# for its tangent.