Let $O_{\theta_1}$ be the single-qubit observable with +1 -eigenvector
$$
\left|v_1\right\rangle=\cos \theta_1|0\rangle+\sin \theta_1|1\rangle
$$
and -1 -eigenvector
$$
\left|v_1^{\perp}\right\rangle=-\sin _1 \theta|0\rangle+\cos \theta_1|1\rangle
$$
Similarly, let $O_{\theta_2}$ be the single-qubit observable with +1 -eigenvector
$$
\left|v_2\right\rangle=\cos \theta_2|0\rangle+\sin \theta_2|1\rangle
$$
and -1 -eigenvector
$$
\left|v_2^{\perp}\right\rangle=-\sin \theta_2|0\rangle+\cos \theta_2|1\rangle
$$
Let $O$ be the two-qubit observable $O_{\theta_1} \otimes O_{\theta_2}$. We consider various measurements on the EPR state $|\psi\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. We are interested in the probability that the measurements $O_{\theta_1} \otimes I$ and $I \otimes O_{\theta_2}$, if they were performed on the state $|\psi\rangle$, would agree on the two qubits in that either both qubits are measured in the 1 -eigenspace or both are measured in -1 -eigenspace of their respective single-qubit observables. As in example 4.2.5, we are not interested in the specific outcome of the two measurements, just whether or not they would agree. The observable $O=O_{\theta_1} \otimes O_{\theta_2}$ gives exactly this information.
a. Find the probability that the measurements $O_{\theta_1} \otimes I$ and $I \otimes O_{\theta_2}$, when performed on $|\psi\rangle$, would agree in the sense of both resulting in a +1 eigenvector or both resulting in a -1 eigenvector. (Hint: Use the trigonometric identities $\cos \left(\theta_1-\theta_2\right)=\cos \left(\theta_1\right) \cos \left(\theta_2\right)+\sin \left(\theta_1\right) \sin \left(\theta_2\right)$ and $\sin \left(\theta_1-\theta_2\right)=\sin \left(\theta_1\right) \cos \left(\theta_2\right)-\cos \left(\theta_1\right) \sin \left(\theta_2\right)$ to obtain a simple form for your answer.)
b. For what values of $\theta_1$ and $\theta_2$ do the results always agree?
c. For what values of $\theta_1$ and $\theta_2$ do the results never agree?
d. For what values of $\theta_1$ and $\theta_2$ do the results agree half the time?
e. Show that whenever $\theta_1 \neq \theta_2$ and $\theta_1$ and $\theta_2$ are chosen from $\left\{-60^{\circ}, 0^{\circ}, 60^{\circ}\right\}$, then the results agree $1 / 4$ of the time and disagree $3 / 4$ of the time.