The present discussion arises from [this MO question][1]. Below, $e$ stands for [Euler's number][2] 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)][3] 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 $\xi_t$ such that (1) holds? **Example:** $\xi_1=\tau$.


[1]: https://mathoverflow.net/questions/354655/how-to-prove-ex-left-int-xx1-sinet-mathrm-d-t-right-le-1-4
[2]: https://www.mathsisfun.com/numbers/e-eulers-number.html
[3]: https://en.wikipedia.org/wiki/Mean_value_theorem