2.8.7 $\arctan x$ 与 $\arctan y$ 的和与差

$$\begin{array}{}\text{(2.159a)}& \arctan x+\arctan y=\arctan \frac{x+y}{1-xy}\left(xy<1\right)\end{array}$$$$\begin{array}{}\text{(2.159b)}& =\pi +\arctan \frac{x+y}{1-xy}\left(x>0,xy>1\right)\end{array}$$$$\begin{array}{}\text{(2.159c)}& =-\pi +\arctan \frac{x+y}{1-xy}\left(x<0,xy>1\right).\end{array}$$$$\begin{array}{}\text{(2.160a)}& \arctan x-\arctan y=\arctan \frac{x-y}{1+xy}\left(xy>-1\right)\end{array}$$$$\begin{array}{}\text{(2.160b)}& =\pi +\arctan \frac{x-y}{1+xy}\left(x>0,xy<-1\right)\end{array}$$$$\begin{array}{}\text{(2.160c)}& =-\pi +\arctan \frac{x-y}{1+xy}\left(x<0,xy<-1\right)\end{array}$$