How to show that this real function is not periodic? 
How can one prove that $$\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$ is not periodic?

This question is motivated by the harmonic spectral representation of time series. Indeed, it is easy to show that a path of a time series given by $$ \cos(\lambda_1 t) + \cos(\lambda_2 t)$$ is periodic if $\frac{\lambda_1}{\lambda_2} \in \mathbb{Q}$. In the above example this is not the case since $\frac{\lambda_1}{\lambda_2} = \pi/2.$
So the case above could serve as an example for the statement: if $\frac{\lambda_1}{\lambda_2} \not\in \mathbb{Q}$ then the path of a harmonic time series is in general not periodic.

The function 
$$f(t)=\cos\left(\frac{\pi}{2} t \right)+\cos\left(t \right)$$
is even; so $f(t+T)=f(t)  \Leftrightarrow f(t-T)=f(t),~~ \forall t$.
Adding the equations
$$f(t+T)=f(t), $$
$$f(t-T)=f(t), $$
we arrive at 
$$\cos t \cos T + \cos\left(\frac{\pi}{2} t \right)\cos\left(\frac{\pi}{2} T \right)=
\cos t+ \cos\left(\frac{\pi}{2} t \right),~~ \forall t. $$
Then, for $t=0$ we have
$$ \cos T + \cos\left(\frac{\pi}{2} T \right)= 2 $$
which implies $T=2m\pi$ and $T=4n$, for $n,m\in\mathbb Z$. In summary, there exists no solution to the periodicity problem ($T=0$ is no period in our conventions).
