What is the greatest value of $\sin x \cos y + \sin y \cos z + \sin z \cos x$, where $x,y,z$ are real numbers?

$ 
\textbf{(A)}\ \sqrt 2
\qquad\textbf{(B)}\ \dfrac 32
\qquad\textbf{(C)}\ \dfrac {\sqrt 3}2
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ 3
$
1. We start by considering the expression \(\sin x \cos y + \sin y \cos z + \sin z \cos x\). We aim to find its maximum value for real numbers \(x, y, z\).

2. We can assume that \(x, y, z\) are between \(0\) and \(\frac{\pi}{2}\). This is because replacing \(x, y, z\) with their reference angles in this range switches all sines and cosines to positive without changing their absolute values, and hence only increases the sum.

3. Without loss of generality (WLOG), we can let \(y\) be the central number of \(x, y, z\). This means that \(y\) is between \(x\) and \(z\) in terms of their values.

4. Consider the sequences \(\sin x, \sin y, \sin z\) and \(\cos z, \cos y, \cos x\). These sequences are sorted similarly because \(\sin\) and \(\cos\) are both non-decreasing functions in the interval \([0, \frac{\pi}{2}]\).

5. By the rearrangement inequality, the sum \(\sin x \cos y + \sin y \cos z + \sin z \cos x\) is maximized when the sequences are similarly sorted. Therefore, we have:
   \[
   \sin x \cos y + \sin y \cos z + \sin z \cos x \leq \sin x \cos z + \sin y \cos y + \sin z \cos x
   \]

6. We know that \(\sin y \cos y = \frac{1}{2} \sin 2y\). Thus, the expression becomes:
   \[
   \sin x \cos z + \frac{1}{2} \sin 2y + \sin z \cos x
   \]

7. The maximum value of \(\sin 2y\) is 1, which occurs when \(y = \frac{\pi}{4}\). Therefore, \(\frac{1}{2} \sin 2y = \frac{1}{2}\).

8. For the remaining terms, \(\sin x \cos z + \sin z \cos x\), we use the identity \(\sin (x+z) = \sin x \cos z + \cos x \sin z\). The maximum value of \(\sin (x+z)\) is 1, which occurs when \(x + z = \frac{\pi}{2}\).

9. Therefore, the maximum value of \(\sin x \cos z + \sin z \cos x\) is 1.

10. Combining these results, we get:
    \[
    \sin x \cos y + \sin y \cos z + \sin z \cos x \leq 1 + \frac{1}{2} = \frac{3}{2}
    \]

11. Equality happens when \(y = \frac{\pi}{4}\) and \(x + z = \frac{\pi}{2}\). Also, we require that \(\cos y = \cos z\) or \(\sin x = \sin y\) for the rearrangement inequality to be equality, and hence \(x = y = z = \frac{\pi}{4}\).

Therefore, the greatest value of \(\sin x \cos y + \sin y \cos z + \sin z \cos x\) is \(\boxed{\frac{3}{2}}\).