For non-negative $a$ and $b$ with $a+b \leq c$ for a small constant $c$, what is the minimum of $\cos a + \cos b$? 
Let $a,b \geq 0$ with $a+b \leq c$ for a small constant $c$ between $0$ and $1$.
What is the minimum of $\cos(a) + \cos(b)$?

I conjecture it is $\cos(0)+\cos(c) = 1 + \cos(c)$ but I have no proof for this.

This is not an answer but it strengthen your guess. Suppose $a+b=2k\leq c$ and let $a=k-\varepsilon$ and $b=k+\varepsilon$ then:
$$f(a,b) :=\cos(a)+\cos(b)=2\cos(k)\cos(\varepsilon),$$
Now one can see that when $\varepsilon\to 0$, $f(a,b)$ increases and for $\varepsilon\to k$, $f(a,b)$ decreases. So probably the minimum occur when $\varepsilon=k$ and $k\to c/2$; i.e. $a=0$ and $b=c$.
