If $n$ is an integer, $n \cdot 180^{\circ}$ represents an integer multiple of $180^{\circ}$ and $(2 n+1) \cdot 90^{\circ}$ represents an odd integer multiple of $90^{\circ}$, and so on. Determine whether the expression is equal to $0,1,-1$, or is undefined.
\[
\sec \left[(2 n+1) \cdot 180^{\circ}\right]
\]
\[
\left.\sec \left[(2 n+1) \cdot 180^{\circ}\right]\right]
\]

Step 1 ：First, we need to understand the meaning of the problem. The problem is asking us to determine the value of the secant function at an odd multiple of 180 degrees.

Step 2 ：The secant function is the reciprocal of the cosine function, so we can rewrite the expression as \(1 / \cos[(2n+1) \cdot 180^{\circ}]\).

Step 3 ：The cosine function has a period of 360 degrees, so \(\cos[(2n+1) \cdot 180^{\circ}]\) is equivalent to \(\cos[180^{\circ}]\).

Step 4 ：The value of \(\cos[180^{\circ}]\) is -1.

Step 5 ：Therefore, the expression \(1 / \cos[(2n+1) \cdot 180^{\circ}]\) is equivalent to \(1 / -1\), which simplifies to -1.

Step 6 ：So, the value of \(\sec[(2n+1) \cdot 180^{\circ}]\) is \(-1\).

Step 7 ：Finally, we check our result. The secant function is undefined at multiples of 180 degrees, but the problem specifies that we are considering odd multiples of 180 degrees. At these points, the secant function is defined and equal to -1, so our result is correct.

Step 8 ：\(\boxed{-1}\) is the final answer.