Applying the MVT we know there exists a constant $c \in (a,b)$ such that $$ f'(c) = \frac{f(b)- f(a)}{b-a},$$ and $$ f'(X_t) = Cos(X_t)dB_t-\frac{1}{2}Sin(X_t) dt .$$ Hence $$ f'(c) = Cos(c)dB_t-\frac{1}{2}Sin(c) dt, $$ and $$ |Cos(c)dB_t-\frac{1}{2}Sin(c) dt| = \left|\frac{f(b) - f(a)}{b - a}\right|$$
I'm unsure how to show the LHS is bounded by some $k$. My ultimate goal is to show that if
$$|Cos(c)dB_t-\frac{1}{2}Sin(c) dt|\le k$$
then
$$|f(b)-f(a)|\le k|b-a|$$
and we would then be able to deduce it is Lipschitz.
There is a stochastic analogue of the mean-value theorem See this paper.
To summarize, it states that if $X$ and $Y$ are non-negative random variable such that $Pr(X>t)\le Pr(Y>t)$ for all $t\in \mathbb{R}$ and $E[X]<E[Y]<\infty$, and $g$ is a measurable and differentiable function such that $E[g(X)]$ and $E[g(Y)]$ are finite, and $g'$ is measurable and Riemann integrable on $[x,y]$ for all $y\ge x\ge 0$, then there exists a random variable $Z$ such that
$$E[g(X)]-E[g(Y)]=E[g'(Z)]\,(E[X]-E[Y])$$