Answer
$x=\frac{\pi}{4}+\pi k$
Work Step by Step
First, note that to have a horizontal tangent, we must have a slope of zero. Since the derivative of a function gives us the slope at that point we will determine when $f'(x)=0$.
If $f(x)=\mathrm{e}^{x}\cos{x}$ then $f'(x)=\mathrm{e}^{x}\cos{x}-\mathrm{e}^{x}\sin{x}=\mathrm{e}^{x}(\cos{x}-\sin{x})$ using the product rule.
$f'(x)=\mathrm{e}^{x}(\cos{x}-\sin{x})=0 \Rightarrow \mathrm{e}^{x}=0$ or $\cos{x}-\sin{x}=0$. Since $\mathrm{e}^{x}\ne0$ we will solve
$\cos{x}-\sin{x}=0$.
If $\cos{x}-\sin{x}=0 \Rightarrow \sin{x}=\cos{x} \Rightarrow \tan{x}=1$ (dividing both sides by $\cos{x}$). Thus, $x=\frac{\pi}{4}+2\pi k, \frac{5\pi}{4}+2\pi k$ or simply $\frac{\pi}{4}+\pi k$.
