`A=[(1,tan x),(-tan x,1)]` and `f(x)` is defined as `f(x)=` det. `(A^(T)A^(-1))` then the value of `ubrace(f(f(f(f..........f(x)))))_("n times")` is `(n ge 2)`_______ .

Answer

2

Explanation

`because A = [[1,tan x],[-tan x,1]]`
`therefore det A + [[1, tan x],[-tan x, 1]]= (1+ tan^(2)x) = ""^(2)x`
`rArr det A^(T) = det A = sec^(2) x `
Now, `f(x) = det (A^(T) A^(-1)) = (det A^(T)) (detA^(-1))`
`= ( det A^(T) ) (det A)^(-1) = (det(A^(T)))/(detA)= 1 `
`therefore underset("n times")(underbrace(lambda = f(f(f(f...f(x))))))=1 [because f(x) = 1]`
Hence, `2^(lambda)= 2^(1) = 2`