Question 159
Let $\theta_i \in (-\frac{\pi}{2}, \frac{\pi}{2})$, $i = 1, 2, 3, 4$. Prove that there exists $x \in \mathbb{R}$ such that the following two inequalities
$$
\cos^2\theta_1 \cos^2\theta_2 - (\sin\theta_1 \sin\theta_2 - x)^2 \ge 0, \quad ①
$$
$$
\cos^2\theta_3 \cos^2\theta_4 - (\sin\theta_3 \sin\theta_4 - x)^2 \ge 0, \quad ②
$$
hold simultaneously if and only if
$$
\sum_{i=1}^{4} \sin^2 \theta_i \le 2 \left( 1 + \prod_{i=1}^{4} \sin \theta_i + \prod_{i=1}^{4} \cos \theta_i \right). \quad ③
$$
$$
\cos^2\theta_1 \cos^2\theta_2 - (\sin\theta_1 \sin\theta_2 - x)^2 \ge 0, \quad ①
$$
$$
\cos^2\theta_3 \cos^2\theta_4 - (\sin\theta_3 \sin\theta_4 - x)^2 \ge 0, \quad ②
$$
hold simultaneously if and only if
$$
\sum_{i=1}^{4} \sin^2 \theta_i \le 2 \left( 1 + \prod_{i=1}^{4} \sin \theta_i + \prod_{i=1}^{4} \cos \theta_i \right). \quad ③
$$