That's a modified exercise taken from a admission test to a university.

Let $x$, $y$ and $\theta$ real numbers such that $$\left \{ \begin{array}{l} x\sin \theta + y \cos \theta = 2a \sin 2\theta\\ x \cos \theta - y \sin \theta = a \cos \theta\\ \end{array} \right.$$ where $a$ is a real constant. How can we determine two real constants $\mu$ ($\mu>0$) and $k$ such that $$(x+y)^\mu+(x-y)^\mu=k$$ without using brute force?


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