Is this a correct method for finding Fourier series coefficients?

Question

I'm trying to find Fourier series coefficients $c_n$ for given signal

$x(t)=\cos(2\pi t)+\cos(4\pi t)$

Solution: $$x(t)=\sum_{n=-\infty}^\infty c_n e^{j \frac{2\pi}{T_0}nt}$$ $$\cos(2\pi t)+\cos(4\pi t)=\sum_{n=-\infty}^\infty c_n e^{j \frac{2\pi}{T_0}nt}$$ $$e^{j \frac{2\pi}{T_0}nt}=\cos(2\pi nt)+j\sin(2\pi nt)$$ The signal is periodic in $t$ with period $1$ $$cos(2\pi t)+cos(4\pi t)=\sum_{n=-\infty}^\infty c_n (\cos(2\pi nt)+j\sin(2\pi nt))$$ $$\cos(2\pi t)+\cos(4\pi t)=...+c_{-2}(\cos(4\pi nt)-j\sin(4\pi nt))+c_{-1}(\cos(2\pi nt)-j\sin(2\pi nt))+c_{0}(\cos(0)+j\sin(0))+c_{1}(\cos(2\pi nt)+j\sin(2\pi nt))+c_{2}(\cos(4\pi nt)+j\sin(4\pi nt))+...$$ $$\cos(2\pi t)+\cos(4\pi t)=...+\cos(4\pi t)(c_{-2}+c_{2})+j\sin(4\pi t)(c_{2}-c_{-2})+\cos(2\pi t)(c_{-1}+c_{1})+j\sin(2\pi t)(c_{1}-c_{-1})+c_0+...$$ $$c_n= \begin{cases} 1/2, & \text{for $n$= -2, -1, 1, 2} \\ 0, & \text{other} \end{cases}$$

Is this a correct solution?

Answer 1

Yes. Nevertheless, it's too long. Instead, directly replace into expression:

$$x(t)=\cos(2\pi t)+\cos(4\pi t)$$

the cosines by their expression with Euler formulas :

$$\cos 2 \pi t= \tfrac12 (e^{2i \pi t}+e^{-2i \pi t})$$

$$\cos 4\pi t= \tfrac12 (e^{4i \pi t}+e^{-4i \pi t})$$