Question Find the value of $\sqrt{x^{2}-1}$ when $x=\sec \theta$ .

Studdy Solution

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.

STEP 8

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

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