A little too complex

$\begin{aligned} S & = \sum_{n=0}^\infty \frac {\cos (nx)}{3^n} \\ & = \sum_{n=0}^\infty \frac {{\color{#3D99F6}\Re} \left \{e^{nxi} \right \}}{3^n} \quad \quad \small \color{#3D99F6} \Re \{z\} \text{ denotes the real part of }z \\ & = \Re \left \{ \sum_{n=0}^\infty \frac {e^{nxi} }{3^n} \right \} \\ & = \Re \left \{\frac 1{1-\frac {e^{xi}}3} \right \} \quad \quad \small \color{#3D99F6} \text{for } \left|\frac {e^{xi}}3\right| = \frac 13 < 1 \\ & = \Re \left \{\frac 3{3-e^{xi}} \right \} \\ & = \Re \left \{\frac 3{3- \cos x - i \sin x} \right \} \\ & = \Re \left \{\frac {3(3- \cos x + i \sin x)}{(3- \cos x - i \sin x)(3- \cos x + i \sin x)} \right \} \\ & = \Re \left \{\frac {3(3- \cos x + i \sin x)}{(3- \cos x)^2 + \sin^2 x} \right \} \\ & = \Re \left \{\frac {9- 3\cos x + i3 \sin x}{9- 6\cos x + \cos^2 x + \sin^2 x} \right \} \\ & = \Re \left \{\frac {9- 3\cos x + i3 \sin x}{10- 6\cos x} \right \} \\ & = \frac {9- 3\cos x }{10- 6\cos x} = \frac {9- 3 \cdot \frac 13 }{10- 6\cdot \frac 13} = \boxed{1} \end{aligned}$

Let

$\begin{aligned} S_{1} & = \dfrac{\cos{0}}{3^{0}}+\dfrac{\cos{x}}{3^{1}}+\dfrac{\cos{(2x)}}{3^{2}}+\cdots \\ S_{2} & = \dfrac{\sin{0}}{3^{0}}+\dfrac{\sin{x}}{3^{1}}+\dfrac{\sin{(2x)}}{3^{2}}+\cdots \\ iS_{2} & = i\dfrac{\sin{0}}{3^{0}}+i\dfrac{\sin{x}}{3^{1}}+i\dfrac{\sin{(2x)}}{3^{2}}+\cdots \end{aligned}$

Add both the equations .

$\begin{aligned} S_{1}+iS_{2} & = \left( \dfrac{\cos{0}}{3^{0}}+i\dfrac{\sin{0}}{3^{0}} \right) + \left( \dfrac{\cos{x}}{3^{1}}+i\dfrac{\sin{x}}{3^{1}} \right) + \left( \dfrac{\cos{(2x)}}{3^{2}}+i\dfrac{\sin{(2x)}}{3^{2}} \right) + \cdots \\ & = \dfrac{1}{3^{0}}e^{0\cdot i}+ \dfrac{1}{3^{1}}e^{x\cdot i}+ \dfrac{1}{3^{2}}e^{2x\cdot i}+\cdots \\ & = \dfrac{1}{1-\dfrac{e^{ix}}{3}} \end{aligned}$

$e^{ix} = \cos{x}+i\sin{x} = \dfrac{1}{3} + i \dfrac{k}{3} \hspace{8mm} \text{Let } \sin{x} = \dfrac{k}{3} \hspace{2mm} , k = \pm 2\sqrt{2}$

$S_{1}+iS_{2} = \dfrac{1}{1-\frac{1}{9}(1+ik)} = \dfrac{9}{8-ik} \\ S_{1}+iS_{2} = \dfrac{9(8+ik)}{64+k^2 } \\ S_{1}+iS_{2} = \dfrac{72}{64+k^2 } +i\left(\dfrac{9k}{64+k^2}\right) \\ \boxed{S_{1} = \dfrac{72}{72} = 1} \hspace{5mm} S_{2} = \pm\dfrac{\sqrt{2}}{4}$

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It is kind of odd for

$\dfrac{\cos{x}}{3^{1}}+\dfrac{\cos{(2x)}}{3^{2}}+\dfrac{\cos{(3x)}}{3^{3}}+\cdots = 0 \hspace{5.5mm}$ at $\cos x =\dfrac{1}{3}$