Answer
The values of $x$ where the tangent line is horizontal are
$x=\dfrac{\pi}{4},\dfrac{3\pi}{4},\dfrac{5\pi}{4},\dfrac{7\pi}{4}$
Work Step by Step
The value of the derivative is the slope of the tangent line.
Since the tangent line is horizontal, so the slope of the tangent will be zero.
Since slope and derivative are the same.
Derivative of $y = \sin {x} \cos {x}$ will also be zero.
That is, $\dfrac{dy}{dx}=\dfrac{d}{dx}\left(\sin{x}\cos{x}\right)=0$
Now use the product rule to solve for x.
$\dfrac{d}{dx}\left(\sin{x}\right)\cos{x}+\dfrac{d}{dx}\left(\cos{x}\right)\sin{x}=0$
$\implies \cos{x}\times\cos{x}+(-\sin{x})\sin{x}=0$
$\implies \cos^{2}{x}-\sin^{2}{x}=0$
Now factorize and solve.
$(\cos{x}+\sin{x})(\cos{x}-\sin{x})=0$
$\implies (\cos{x}+\sin{x})=0$ or $(\cos{x}-\sin{x})=0$
$\implies \cos{x}=-\sin{x}$ or $\cos{x}=\sin{x}$
Now divide by $\cos{x}$ in both the equation.
$\implies \dfrac{-\sin{x}}{\cos{x}}=1$ or $\dfrac{\sin{x}}{\cos{x}}=1$
$\implies \tan{x}=-1$ or $\tan{x}=1$
Since, $x$ is between $0$ and $2\pi$.
If $\tan{x}=1$
Then, $x=\dfrac{\pi}{4},\dfrac{5\pi}{4}$
If $\tan{x}=-1$
Then, $x=\dfrac{3\pi}{4},\dfrac{7\pi}{4}$