The maximum value of $\sin x \cdot \cos x$ is:

To find the maximum value of $\sin x \cdot \cos x$, use the double angle formula.

The double angle formula states:

$\sin(2x) = 2\sin x \cos x$

Rearranging:

$\sin x \cos x = \frac{\sin(2x)}{2}$

The sine function has a range of $[-1, 1]$.

Maximum value of $\sin(2x) = 1$

Therefore:

Maximum of $\sin x \cos x = \frac{1}{2}$

This maximum occurs when $\sin(2x) = 1$, which happens when $2x = \frac{\pi}{2}$, giving $x = \frac{\pi}{4}$.

At $x = \frac{\pi}{4}$:

$\sin\left(\frac{\pi}{4}\right) \cdot \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2}$

The maximum value of $\sin x \cdot \cos x$ is $\frac{1}{2}$.