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I am trying to find all $\theta\in\mathbb{R}$ such that $$\cos(t\cos\theta)\cos(t\sin\theta) \sin 2\theta = \sin(t\cos\theta)\sin(t\sin\theta)$$ holds for all $t\in\mathbb{R}$. Just by looking at the equation it is clear that $\theta = \frac{\pi n}{2}, n \in\mathbb{Z},$ are solutions.

I think that there are no more solutions, since if we assume that $\theta \not= \frac{\pi n}{2},$ we can rearrange the equation to get $$ \sin 2\theta = \frac{\sin(t\cos\theta)\sin(t\sin\theta)}{\cos(t\cos\theta)\cos(t\sin\theta)} = \tan(t\cos\theta)\tan(t\sin\theta). $$ Is it safe to conclude from here that the RHS will in general depend on $t$, so no single choice of $\theta$ will work for all $t$? Is there a better way of seeing this?

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