Starting from the neutrino flavor eigenstates in a 2 -flavor model with vacuum mixing angle $\theta$,
$$
\begin{aligned}
& \left|v_{\mathrm{e}}\right\rangle=\cos \theta\left|v_1\right\rangle+\sin \theta\left|v_2\right\rangle \\
& \left|v_\mu\right\rangle=-\sin \theta\left|v_1\right\rangle+\cos \theta\left|v_2\right\rangle,
\end{aligned}
$$
show that the mass eigenstates may be expressed as $$
\begin{aligned}
& \left|v_1\right\rangle=\cos \theta\left|v_{\mathrm{e}}\right\rangle-\sin \theta\left|v_\mu\right\rangle \\
& \left|v_2\right\rangle=\sin \theta\left|v_{\mathrm{e}}\right\rangle+\cos \theta\left|v_\mu\right\rangle,
\end{aligned}
$$
so that the time-evolved neutrino state
$$
|v(t)\rangle=\cos \theta e^{-i E_1 t}\left|v_1(0)\right\rangle+\sin \theta e^{-i E_2 t}\left|v_2(0)\right\rangle
$$
may be written as the mixed-flavor state
$$
\begin{aligned}
|v(t)\rangle= & \left(\cos ^2 \theta e^{-i E_1 t}+\sin ^2 \theta e^{-i E_2 t}\right)\left|v_{\mathrm{c}}\right\rangle \\
& +\sin \theta \cos \theta\left(-e^{-i E_1 t}+e^{-i E_2 t}\right)\left|v_\mu\right\rangle .
\end{aligned}
$$
Use this result to show that the probabilities to detect a $v_{\mathrm{e}}$ or $v_\mu$ after a time $t$ are given by
$$
\begin{gathered}
P\left(v_{\mathrm{c}} \rightarrow v_{\mathrm{e}}, t\right)=1-\frac{1}{2} \sin ^2(2 \theta)\left[1-\cos \left(E_2-E_1\right) t\right], \\
P\left(v_{\mathrm{e}} \rightarrow v_\mu, t\right)=\frac{1}{2} \sin ^2(2 \theta)\left[1-\cos \left(E_2-E_1\right) t\right],
\end{gathered}
$$
respectively.***