The present discussion arises from this MO question. Below, $e$ stands for Euler's number and let $$\tau:=\arccos\left(\frac{\sin e-\sin 1}{e-1}\right)\approx 1.82\cdots.$$

An application of the Mean Value Theorem (for derivatives) to the function $f(t)=\sin t$ leads to $$\frac{\sin(e\,t)-\sin(t)}{e\,t-t}=\cos(\xi_tt) \qquad \text{for some $1\leq\xi_t\leq e$}. \tag1$$

QUESTION. Is it true that for each $t>0$, one can always find some $\xi_t\geq\tau$ such that (1) holds? Example: $\xi_1=\tau$.