MHB Proving Identity: Tan(Tan⁻¹(x) + Tan⁻¹(y))

[karush]

Prove identity

$$\tan[\tan^{-1} (x) +\tan^{-1} (y)] =(x+y)(x-y)$$

Since $$\tan\left({\tan^{-1} \left({a}\right)}\right)=a$$

And by sum formula of $\tan{(x+y)}$ then $$=\frac{x+y}{1-xy}$$

But then?

 

[I like Serena]

Prove identity

$$\tan[\tan^{-1} (x) +\tan^{-1} (y)] =(x+y)(x-y)$$

Since $$\tan\left({\tan^{-1} \left({a}\right)}\right)=a$$

And by sum formula of $\tan{(x+y)}$ then $$=\frac{x+y}{1-xy}$$

But then?

Hey karush,

It's not true.
Pick for instance $x=y=\tan(\pi/6)$, then $\tan(\pi/6+\pi/6)\ne 0$.

 

[karush]

OK, think I see why, should be $$\frac{x+y}{1-xy}$$ in original op
Was reading a very hazy cell phone pic