How to prove $cos(t) + cos(\pi t)$ is non periodic function? Also can I represent this signal using fourier series? I would just want to prove $\cos(t) + \cos(\pi t)$ is non periodic. 

I don't know where to start it. Also I know that individually these signals ie $\cos(t)$ and $\cos(\pi t)$ are periodic with frequencies $1/(2 \pi)$ Hz and $1/2$ Hz respectively. But since the superpositon of these signals is not periodic should we represent them using Fourier transform only? What is the harm in drawing the frequency components at $1/(2 \pi)$ Hz and $1/2$ Hz using fourier series?
$cos(t)$ and $cos(\pi t)$ are both periodic with periods $2\pi$ and $2$ respectively. To find the period of the sum we need to find an integers $n,m$ such that $\pi n = 2 m$ or $\pi / 2 = m /n $, which is not possible since $\pi$ is irrational.

I think the Fourier series uses specific frequencies to form the signal. You can't necessarily put a frequency component at irrational frequencies.

Fourier series work fine with irrational frequencies too, but the frequencies are all harmonics, that is, integer multiples, of the fundamental frequency.