<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] Apply $\;\tan\;$ to both sides and use trigonometric identities:
$$\begin{align*}\bullet&\tan (\arctan x+\arctan y)=\frac{\tan\arctan x+\tan\arctan y}{1-\tan\arctan x\tan\arctan y}=\frac{x+y}{1-xy}\\{}\\\bullet&\tan\left(\pi+\arctan\frac{x+y}{1-xy}\right)=\tan\arctan\frac{x+y}{1-xy}=\frac{x+y}{1-xy}\end{align*}$$
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