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Is it possible to have a function $f(x)$ that has the same Maclaurin series as $\sin(x)$ with $f(x)\ne \sin(x)$ for $x\ne 0$?

At first glance, nothing seems impossible about this. But I would need that $f^{(k)}(0)=\sin^{(k)}(0)$ for $k=0,1,2,\ldots$ and that $f(x)\ne\sin(x)$ for $x\ne0$, which I can't find a way to construct.

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    Add any non-zero [flat function](https://en.wikipedia.org/wiki/Flat_function) to $\sin x$. (Clarification: It only has to be flat at $x = 0$, equal to $0$ there, and should be non-zero everywhere else.) – 2017-01-30
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    That'll do it. Thanks! – 2017-01-30

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