I am trying to work through a small problem (finding a Fréchet Derivative), and I arrive at a function that is "obviously" less than $x^2$ for all $x$, which would be very nice to prove. I say "obviously" because by looking at the plots I visually notice that it is always smaller than $x^2$. But I cannot figure out how to show it rigorously.
Here is the function, with $a$ some constant real number.
$$f_a(x) = |\sin(a+x)-\sin(a)-\cos(a)x|$$
I have tried to rewrite the expression with some trigonometric identities, like so: $$f_a(x) = |2\sin(x/2)\cos(a + x/2) - \cos(a)x|$$ But this gets me nowhere. The fact that the whole expression is inside absolute values means I cannot really do much with it, unless I use the triangle inequality to get $$f_a(x) \leq |2\sin(x/2)\cos(a + x/2)| + |\cos(a)x|$$but this does not seem strict enough to show the upper limit of $x^2$.
How would I go about showing something like this?