Real Signal Properties and $\cos(t)$ Fourier coefficients

Question

I am struggling to conceptually understand why the Fourier coefficients for $\cos(t)$ are $a_1 = 1/2$ and $a_{-1} = 1/2$ in light of the fact that, for a real and even signal, the Fourier coefficients should be real and even (I am referring to a complex Fourier series). What am I missing to explain this discrepancy?

Thank you!

Answer 1

$1/2$ is real so these coefficients are real. (All the coefficients you didn't list are $0$, which is also real.) Also, $a_n = a_{-n}$ for all $n$, so this set of coefficients has even symmetry. There is no discrepancy, so I can't guess what you are missing.