$\begin{gathered}\text { We have, } \quad \begin{array}{c}\sin \theta \cosh \alpha=\tan x \\ \cos \theta \sinh \alpha=\sec x\end{array} \\ \cos ^2 \theta \sin ^2 h \alpha-\sin ^2 \theta \cos ^2 h \alpha=\sec ^2 x-\tan ^2 x \\ \cos ^2 \theta \sin ^2 h \alpha-\sin ^2 \theta \cos ^2 h \alpha=1 \\ \cos ^2 \theta \sin ^2 h \alpha-\cos ^2 h \alpha+\cos ^2 \theta \cos ^2 h \alpha=1 \\ \cos ^2 \theta\left(\sin ^2 h \alpha+\cos ^2 h \alpha\right)-\cos ^2 h \alpha=1\end{gathered}$
$\begin{aligned} & \cos ^2 \theta \cosh 2 \alpha-\cos ^2 h \alpha=1 \\ & \cos ^2 \theta \cosh 2 \alpha-\left(\frac{\cosh 2 \alpha+1}{2}\right)=1 \\ & 2 \cos ^2 \theta \cosh 2 \alpha-\cosh 2 \alpha-1=2 \\ & \cosh 2 \alpha\left(2 \cos ^2 \theta-1\right)=3 \Rightarrow \cos 2 \theta \cosh 2 \alpha=3\end{aligned}$