<s>[INST] Eliminate $\theta$ and prove $x^2+y^2=1$ We have:$${ \begin{cases}{2x=y\tan\theta+\sin\theta} \\ {2y=x\cot\theta+\cos\theta}\end{cases} }$$
And want to prove $x^2+y^2=1$
My works:
I multiplied first equation by $\cos\theta$ and second one by $\sin\theta$ and get:
$${ \begin{cases}{2x\cos\theta=y\sin\theta+\sin\theta\cos\theta} \\ {2y\sin\theta=x\cos\theta+\sin\theta\cos\theta}\end{cases} }$$
By extracting $\sin\theta\cos\theta$ we get: $$2x\cos\theta-y\sin\theta=2y\sin\theta-x\cos\theta$$
$$x\cos\theta=y\sin\theta$$
But I don't know whether this helps or not.
 [/INST] You are almost there. You already have $$2x\cos\theta=y\sin\theta+\sin\theta\cos\theta \tag1$$
and $$x\cos \theta = y \sin \theta \tag 2$$
Then $$x\cos \theta = y \sin \theta = 2x\cos \theta - y \sin \theta = \sin \theta \cos \theta$$
and you know $\sin \theta \ne 0, \cos \theta \ne 0$. Can you end it now?
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