Curl of vector field must be zero or not must be zero?

I have to find is it possible for F=sinx i+cosy j+sinxyz k to be the curl of vector field?

Therefore, $\begin{pmatrix}i&j&k\\ \frac{d}{dx}&\frac{d}{dy}&\frac{d}{dz}\\ sinx&cosy&sinxyz\end{pmatrix}$
$(\frac{dsinxyz}{dy}-\frac{dcosy}{dz})i+(\frac{dsinx}{dz}-\frac{dsinxyz}{dx})+(\frac{dcosy}{dx}-\frac{dsinx}{dy})$
=cos(xyz)xz-cos(xyz)yz (smell of calculation mistake

And there are irrotational (div gradF = 0)and rotational vector (div gradF not = 0)

Therefore, my answer is IT'S POSSIBLE, it is rotational curl of vector field.

But that's mean any answer can be curl of vector field...?

Or I misunderstood the question? Or the curl of vector field must equal to 0?

The question seeks you to find if $ \vec F = (\sin x, \cos y,\sin (xyz))$ is curl of another vector field. Say it is curl of a vector field $\vec G$. Then,

$\vec F = \nabla \times \vec G$ and we must have,

$\nabla \cdot \vec F = \nabla \cdot (\nabla \times \vec G) = 0$

But $~ \nabla \cdot \vec F = \cos x - \sin y + xy \cos (xyz) \ne 0$.

So $\vec F$ is not curl of another vector field.