If $A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$, then the value of $\left|A^T A^{-1}\right|$ is

1

$A^{T}=\left[\begin{array}{cc}1 & -\tan x \\ \tan x & 1\end{array}\right]$

$A^{-1}=\frac{1}{1+\tan ^2 x}\left[\begin{array}{cc}1 & -\tan x \\ \tan x & 1\end{array}\right]$

$A^{T} A^{-1}=\left[\begin{array}{cc}\cos 2 x & -\sin 2 x \\ \sin 2 x & \cos 2 x\end{array}\right]$

$\left|A^{T} A^{-1}\right|=1$

Hence (4) is the correct answer.