How do you prove the following trig identity: #tan(x + 45°) - tan(45° - x) ≡ 2tan2x#?

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Answer 1 · 2015-04-15T18:32:52

Use #tan(a+b) = (tana+tanb)/(1-tanatanb)#

and #tan(a-b) = (tana-tanb)/(1+tanatanb)#

and #tan (2a) = tan(a+a) = (2tana)/(1-tan^2a)#

And also use: #tan (pi/4) =1#.

Other than that it's just algebra.

Once you get to

#(tanx + 1)/(1-tanx)-(1-tanx)/(1+tanx)#

you'll want a common denominator, so you get

#((tanx+1)^2-(1-tanx)^2)/(1-tan^2x)#

Now do the algebra to get:

#(4tanx)/(1-tan^2x) = 2( (2tanx)/(1-tan^2x)) = 2 tan2x#