$\mathrm{A}=\left[\begin{array}{cc}1& \tan \mathrm{x}\\ -\tan \mathrm{x}& 1\end{array}\right]$ and f(x) is defined as f(x) = det. $\left({\mathrm{A}}^{\mathrm{T}}{\mathrm{A}}^{-1}\right)$ then the value of $\underset{\mathrm{n}\text{ times }}{\underbrace{\mathrm{f}\left(\mathrm{f}\right(\mathrm{f}(\mathrm{f}\dots \dots .\mathrm{f}(\mathrm{x}\left)\right)\left)\right)}}$  is $(\mathrm{n}\ge 2)$

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

Hence, detA = sec²x

$\therefore \det {\mathrm{A}}^{\mathrm{T}}={\sec }^2\mathrm{x}$

now $\mathrm{f}\left(\mathrm{x}\right)=\det \cdot \left({\mathrm{A}}^{\mathrm{T}}{\mathrm{A}}^{-1}\right)$

$\begin{array}{c}=\left(\det \cdot {\mathrm{A}}^{\mathrm{T}}\right)\left(\det \cdot {\mathrm{A}}^{-1}\right)=\left(\det \cdot {\mathrm{A}}^{\mathrm{T}}\right)(\det \cdot \mathrm{A}{)}^{-1}\\ =\frac{\det \cdot \left({\mathrm{A}}^{\mathrm{T}}\right)}{\det \cdot \left(\mathrm{A}\right)}=1\end{array}$

Hence, f(x) =1