Verify the identity:

tan(2tan-1x)= 2 tan(tan-1x + tan-1x3)

Let tan-1x = A and let tan-1x3 = B

Then tanA = x and tanB = x3

and the identity to verify becomes:

tan(2A)= 2tan(A + B)
                                                2tanA
On the left side use the identity: tan(2A) = ——————————— 
                                              1 - tan2A

                                                 tanA + tanB
On the right side use the identity: tan(A+B) = ————————————————
                                                1 - tanA·tanB


   2tanA           tanA + tanB   
———————————  = 2·————————————————
 1 - tan²A        1 - tanA·tanB  
 
Since tanA = x and tanB = x3, the above becomes


      2x           x + x3   
   ————————  = 2·——————————
    1 - x2        1 - x·x3  
 
Factor the numerator on the right getting x(1 + x)
Multiply x·x3 getting x4

      2x           x(1 + x2)   
   ————————  = 2·————————————
    1 - x2          1 - x4

or putting the 2 factor on the right in the numerator

      2x        2x(1 + x2)   
   ————————  = ————————————
    1 - x2        1 - x4


Factor the denominator on the right as the
difference of two perfect squares:

      2x           2x(1 + x2)   
   ————————  = ——————————————————
    1 - x2      (1 - x2)(1 + x2)

Cancel the (1 + x2)'s on the right:
                        1
      2x           2x(1 + x2)   
   ————————  = ——————————————————
    1 - x2      (1 - x2)(1 + x2)
                           1
                        
      2x           2x   
   ————————  = ——————————
    1 - x2       1 - x2
                           
Edwin