<s>[INST] Proves of identities in inverse trigonometry Can someone please help me prove the following results from inverse trigonometry?
$$\tan^{-1}x + \tan^{-1}y  = \pi + \tan^{-1}\frac{x+y}{1-xy}( x>0, y>0, xy>1)$$
and
$$\tan^{-1}x + \tan^{-1}y  = -\pi + \tan^{-1}\frac{x+y}{1-xy} ( x<0, y< 0, xy>1)$$
I know the prove for $\tan^{-1}x + \tan^{-1}y  =\tan^{-1}\frac{x+y}{1-xy} (  xy<1)$ but cant prove the other two. Please do help.
Thanks in advance :)
 [/INST] Hint:  Let $\tan^{-1} x = a$ and $\tan^{-1} y = b$. Then $x = \tan a$ and $y = \tan b$.
$$\frac{x+y}{1-xy} = \frac{\tan a + \tan b}{1- \tan a \tan b}$$
$$\frac{x+y}{1-xy} = \tan {(a+b)}$$
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