Is $\sqrt{1-\sin ^2 100^\circ}\cdot \sec 100^\circ = 1$ or $-1$?

Question

The equation will simplify to

\begin{align} & = \sqrt{\cos^2 100^\circ}\cdot \sec100^\circ \\[8pt] & = \cos100^\circ\cdot\sec100^\circ \\[8pt] & = 1 \end{align}

But the answer key says that the correct answer is $-1$?

Answer 1

We know that $\sqrt{x^2}=|x|$. So $\cos100$ will be negative because in the second quadrant. So $\sqrt{\cos^2100}=-\cos100$.

Answer 2

$$\sqrt{x^2}=|x|$$ ${}{}{}{}{}{}{}{}$

Answer 3

Root of $\cos(100)^2$ is always positive. So root of $\cos(100)^2$ is also positive. And the value of root of $1-\sin(100)^2$ will be $+\cos100$. Now $\sec100$ is negative, so the value is $1/-\cos100$. So $\cos100$ and $-\cos100$ get cancelled and we get -1 as the answer.