For \[{\text{A}} = \left[ {\begin{array}{*{20}{c}} 1&{\tan \,{\text{x}}} \\ { – \tan {\text{ x}}}&1 \end{array}} \right],\] the determinant of ATA-1 is A. sec2x B. cos4x C. 1 D. 0

The determinant of ATA-1 is $1$.

To see this, we can use the formula for the determinant of a 2×2 matrix:

$$\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = (a-bd)$$

In this case, we have $a=1$, $b=\tan x$, $c=-\tan x$, and $d=1$. Substituting these values into the formula, we get:

$$\det \begin{bmatrix} 1 & \tan x \\ -\tan x & 1 \end{bmatrix} = (1-(-\tan x)\tan x) = 1$$

Therefore, the determinant of ATA-1 is $1$.

Here is a brief explanation of each option:

• Option A: sec2x. This is not the correct answer because sec2x is not equal to 1.
• Option B: cos4x. This is not the correct answer because cos4x is not equal to 1.
• Option C: 1. This is the correct answer because the determinant of ATA-1 is 1.
• Option D: 0. This is not the correct answer because the determinant of ATA-1 is not equal to 0.