While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of positive integers: $$\begin{align} \arctan\varphi&=2\arctan1-\frac12\,\arctan2\\ \arctan\varphi^3&=\arctan1+\frac12\,\arctan2\\ \arctan\varphi^5&=4\arctan1-\frac32\,\arctan2\\ \arctan\varphi^7&=3\arctan1+\frac12\,\arctan2-\arctan5\\ \arctan\varphi^9&=\arctan1-\frac12\,\arctan2+\arctan4\\ \arctan\varphi^{11}&=5\arctan1+\frac12\,\arctan2-\arctan5-\arctan34\\ \arctan\varphi^{13}&=3\arctan1-\frac12\,\arctan2+\arctan4-\arctan89\\ \arctan\varphi^{15}&=-2\arctan1+\frac32\,\arctan2+\arctan11 \end{align}$$ I was not able to find such a representation for $\arctan\varphi^{17}$ though.

Question 1.Can we prove that it does not exist?

I also could not find such a representation for any positive even power.

Question 2.Can we prove that it does not exist for any positive even power?

Question 3.Is there a simple way to determine if such a representation exists for a given power?