Certainly, here is the translation of the provided text into English, preserving the original formatting:

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Clearly, equations (1) and (2) are equivalent to
$$\begin{array}{l} \sin \theta_{1} \cdot \sin \theta_{2}-\cos \theta_{1} \cdot \cos \theta_{2} \\ \leqslant x \leqslant \sin \theta_{1} \cdot \sin \theta_{2}+\cos \theta_{1} \cdot \cos \theta_{2}, \\ \sin \theta_{3} \cdot \sin \theta_{4}-\cos \theta_{3} \cdot \cos \theta_{4} \\ \leqslant x \leqslant \sin \theta_{3} \cdot \sin \theta_{4}+\cos \theta_{3} \cdot \cos \theta_{4} . \end{array}$$

It is evident that the necessary and sufficient condition for the existence of $x \in \mathbf{R}$ such that equations (4) and (5) hold simultaneously is
$$\begin{array}{l} \sin \theta_{1} \cdot \sin \theta_{2}+\cos \theta_{1} \cdot \cos \theta_{2}- \\ \sin \theta_{3} \cdot \sin \theta_{4}+\cos \theta_{3} \cdot \cos \theta_{4} \geqslant 0, \\ \sin \theta_{3} \cdot \sin \theta_{4}+\cos \theta_{3} \cdot \cos \theta_{4}- \\ \sin \theta_{1} \cdot \sin \theta_{2}+\cos \theta_{1} \cdot \cos \theta_{2} \geqslant 0 . \end{array}$$

On the other hand, using $\sin ^{2} \alpha=1-\cos ^{2} \alpha$, equation (3) can be transformed into
$$\begin{array}{l} \cos ^{2} \theta_{1} \cdot \cos ^{2} \theta_{2}+2 \cos \theta_{1} \cdot \cos \theta_{2} \cdot \cos \theta_{3} \cdot \cos \theta_{4}+ \\ \cos ^{2} \theta_{3} \cdot \cos ^{2} \theta_{4}-\sin ^{2} \theta \cdot \sin ^{2} \theta_{2}+ \\ 2 \sin \theta_{1} \cdot \sin \theta_{2} \cdot \sin \theta_{3} \cdot \sin \theta_{4}-\sin ^{2} \theta_{3} \cdot \sin ^{2} \theta_{4} \geqslant 0, \end{array}$$

which simplifies to
$$\left(\cos \theta_{1} \cdot \cos \theta_{2}+\cos \theta_{3} \cdot \cos \theta_{4}\right)^{2}- \\ \left(\sin \theta_{1} \cdot \sin \theta_{2}-\sin \theta_{3} \cdot \sin \theta_{4}\right)^{2} \geqslant 0,$$

or equivalently,
$$\begin{array}{l} \left(\sin \theta_{1} \cdot \sin \theta_{2}+\cos \theta_{1} \cdot \cos \theta_{2}-\sin \theta_{3} \cdot \sin \theta_{4}+\right. \\ \left.\cos \theta_{3} \cdot \cos \theta_{4}\right)\left(\sin \theta_{3} \cdot \sin \theta_{4}+\cos \theta_{3} \cdot \cos \theta_{4}-\right. \\ \left.\sin \theta_{1} \cdot \sin \theta_{2}+\cos \theta_{1} \cdot \cos \theta_{2}\right) \geqslant 0 . \end{array}$$

When there exists $x \in \mathbf{R}$ such that equations (4) and (5) hold simultaneously, equations (6) and (7) immediately imply equation (8). Therefore, equation (3) holds.

Conversely, when equation (3), or equivalently equation (8), holds, if equations (6) and (7) do not hold, then
$$\begin{array}{l} \sin \theta_{1} \cdot \sin \theta_{2}+\cos \theta_{1} \cdot \cos \theta_{2}-\sin \theta_{3} \cdot \sin \theta_{4}+ \\ \cos \theta_{3} \cdot \cos \theta_{4}<0 \\ \sin \theta_{3} \cdot \sin \theta_{4}+\cos \theta_{3} \cdot \cos \theta_{4}-\sin \theta_{1} \cdot \sin \theta_{2}+ \\ \cos \theta_{1} \cdot \cos \theta_{2}<0 \end{array}$$

Adding these two inequalities, we get
$$2\left(\cos \theta_{1} \cdot \cos \theta_{2}+\cos \theta_{3} \cdot \cos \theta_{4}\right)<0.$$
This contradicts $\theta_{i} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right), i=1,2,3,4$.
Therefore, equations (6) and (7) must hold simultaneously. Hence, there exists $x \in \mathbf{R}$ such that equations (4) and (5) hold simultaneously.