<s>[INST] showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$ I'm having problem with showing that:
$$\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$$
I would need some help in the right direction
 [/INST] From the Article $240,$ Ex$-5$ of Plane Trigonometry(by Loney),
$$\arctan x+\arctan y=\begin{cases} \arctan\frac{x+y}{1-xy} &\mbox{if } xy<1\\ \pi+\arctan\frac{x+y}{1-xy} & \mbox{if } xy>1\end{cases} $$
$$\implies 2\arctan x=\arctan \frac{2x}{1-x^2}\text{  if }x^2<1$$
Here $\displaystyle x=\frac23$
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