Using #t=tan(theta/2)#, prove that #tan(x/2+pi/4)+tan(x/2-pi/4)=2tanx#?

1 Answer
Sonnhard
May 30, 2018

For the proof we use
#tan(x/2+pi/4)=(1+tan(x/2))/(1-tan(x/2))#
and
#tan(x/2-pi/4)=(tan(x/2)-1)/(tan(x/2)+1)#
#tan(pi/4)=1#

Explanation:

So we get

#((tan(x/2)+1)^2+(tan(x/2)-1)*(1-tan(x/2)))/(1-tan^2(x/2))#
This is
#(tan^2(x/2)+1+2tan(x/2)+tan(x/2)-1-tan^2(x/2)+tan(x/2))/(1-tan^2(x/2))#
and this is

#(4tan(x/2))/(1-tan^2(x/2))=2tan(x)#

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