Question 117920
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Verify the identity:

tan(2tan<sup>-1</sup>x)= 2 tan(tan-1x + tan<sup>-1</sup>x<sup>3</sup>)

Let tan<sup>-1</sup>x = A and let tan<sup>-1</sup>x<sup>3</sup> = B

Then tanA = x and tanB = x<sup>3</sup>

and the identity to verify becomes:

tan(2A)= 2tan(A + B)
                                                2tanA
On the left side use the identity: tan(2A) = ——————————— 
                                              1 - tan<sup>2</sup>A

                                                 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 = x<sup>3</sup>, the above becomes


      2x           x + x<sup>3</sup>   
   ————————  = 2·——————————
    1 - x<sup>2</sup>        1 - x·x<sup>3</sup>  
 
Factor the numerator on the right getting x(1 + x)
Multiply x·x<sup>3</sup> getting x<sup>4</sup>

      2x           x(1 + x<sup>2</sup>)   
   ————————  = 2·————————————
    1 - x<sup>2</sup>          1 - x<sup>4</sup>

or putting the 2 factor on the right in the numerator

      2x        2x(1 + x<sup>2</sup>)   
   ————————  = ————————————
    1 - x<sup>2</sup>        1 - x<sup>4</sup>


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

      2x           2x(1 + x<sup>2</sup>)   
   ————————  = ——————————————————
    1 - x<sup>2</sup>      (1 - x<sup>2</sup>)(1 + x<sup>2</sup>)

Cancel the (1 + x<sup>2</sup>)'s on the right:
                        1
      2x           2x<s>(1 + x<sup>2</sup>)</s>   
   ————————  = ——————————————————
    1 - x<sup>2</sup>      (1 - x<sup>2</sup>)<s>(1 + x<sup>2</sup>)</s>
                           1
                        
      2x           2x   
   ————————  = ——————————
    1 - x<sup>2</sup>       1 - x<sup>2</sup>
                           
Edwin</pre>