Expert: Paul Klarreich - 8/14/2009

Question
Verify the identity.

tan(2 tan^-1 x) - 2 tan (tan^-1 x + tan^-1 x^3

Answer
Questioner: Cherry
Country: Trinidad and Tobago
Category: Advanced Math
Private: No
Subject: Re: Trigonometric Identities
Question: Verify the identity.

tan(2 tan^-1 x) - 2 tan (tan^-1 x + tan^-1 x^3 )
........................................
You are missing an equal sign.  That makes it hard.  But I will get you started:
These are your standard reduction formulas: (You already memorized them, right?)

            tan a + tan b
tan(a + b) = ---------------
            1 - tan a tan b

          2 tan a
tan(2a) = -------------
         1 - tan^2(a)


Now here is the clue to handling inverse trig functions.  Remember that:

arc(something) IS AN ANGLE.

Set:

a = arctan x,    so  tan a = x
b = arctan x^3,  so  tan b = x^3

Then  tan(2 tan^-1 x) = tan (2a)
 
          2 tan a
tan(2a) = ------------
         1 - tan^2(a)
    2x
= -------
 1 - x^2

And  tan (tan^-1 x + tan^-1 x^3 ) =  tan(a + b)

            tan a + tan b
tan(a + b) = ---------------
            1 - tan a tan b
  x + x^3
= ----------
  1 - x^4

Take it from there.