Math Snap

STEP 1

Assumptions
1. We are given the expression $\sqrt{x^{2}-1}$.
2. We are also given that $x = \sec \theta$.
3. We need to simplify the expression using the given substitution.

STEP 2

Substitute $x = \sec \theta$ into the expression $\sqrt{x^{2}-1}$.
$$\sqrt{x^{2}-1} = \sqrt{(\sec \theta)^{2}-1}$$

STEP 3

Recall the trigonometric identity $\sec^2 \theta = 1 + \tan^2 \theta$.

STEP 4

Substitute $\sec^2 \theta$ with $1 + \tan^2 \theta$ in the expression.
$$\sqrt{(\sec \theta)^{2}-1} = \sqrt{1 + \tan^2 \theta - 1}$$

STEP 5

Simplify the expression inside the square root.
$$\sqrt{1 + \tan^2 \theta - 1} = \sqrt{\tan^2 \theta}$$

STEP 6

Take the square root of $\tan^2 \theta$, which is $\tan \theta$.
$$\sqrt{\tan^2 \theta} = |\tan \theta|$$

STEP 7

Since $\sec \theta$ is defined for all $\theta$ except where $\cos \theta = 0$ (i.e., $\theta \neq \frac{\pi}{2} + k\pi$ for any integer $k$), and $\tan \theta$ is defined for all $\theta$ except where $\cos \theta = 0$, we can assume that $\theta$ is not an odd multiple of $\frac{\pi}{2}$. Therefore, $\tan \theta$ will not be undefined.

Conclude that the simplified form of the original expression $\sqrt{x^{2}-1}$, given $x = \sec \theta$, is $|\tan \theta|$.
The simplified expression is $|\tan \theta|$.