`A=[[1,t a n x],[-t a n x,1]] \ a n d \ f(x)` is defined as `f(x)=d e t(A^T A^(-1))` for all x in domain, then the value of `(f(f(f(f....f(x))))_` for `(ngeq2)` is _________.

Answer

1

Explanation

`A=[(1,tan x),(-tan x,1)]`
Hence, det. `A=sec^(2)x`
`:.` det `A^(T)=sec^(2)x`
Now `f(x)=` det. `(A^(T)A^(-1))`
`=("det. "A^(T)) ("det. "A^(-1))`
`=("det. "A^(T)) ("det. "A)^(-1)`
`=("det."(A^(T)))/("det. (A)")=1`
Hence, `f(x)=1`.