# Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational.

Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where $a$ is some constant, are both rational too.

It is known that when $\cos(t)$ and $\sin(t)$ are rational, $$\cos t = \frac{m^2-n^2}{m^2+n^2} \quad ; \quad \sin t = \frac{2m}{m^2+n^2}$$ where $m$ and $n$ are integers.

Does it exist some relation between $\sin(t)$ ($\cos(t)$) and $\sin(at)$ ($\cos(at)$) in terms of $m$ and $n$?