If $A=\left(\begin{smallmatrix} 1 & \tan x\\ -\tan x & 1 \end{smallmatrix}\right)$ and $f(x)=\det(A^TA^{-1})$, then what is $f(f(f(f \cdots f(x))))$? 
Let $$A := \begin{bmatrix} 1 & \tan x\\ -\tan x & 1\end{bmatrix}$$ and $$f(x) := \det(A^TA^{-1})$$ Which of the following can not be the value of $f(f(f(f \cdots f(x))))$?

*

*$f^n(x)$

*$1$

*$f^{n-1}(x)$

*$nf(x)$
where $n \geq 2$.


I found $$f(x)=\frac{\det(A^T)}{\det(A)}=\frac{\det(A)}{\det(A)}=1.$$
But I could not figure out the answer. Please help me.

As you figured out that $f(x) = 1$
1.$f^n(x) = 1$ as $f(x) = 1 \Rightarrow 1^n = 1$
2.Use the same argument as $(1)$
3.That is indeed right.
4.$n f(x) = n (1) = n$, which can't be the value of $f(f(f(f(f(....f(x)))))))$
Hence, option $(4)$ is the answer. 
