Let's begin by substituting [tex]\( x = 2 \sec(\theta) \)[/tex] into the expression [tex]\( \sqrt{25 x^2 - 100} \)[/tex].
First, substitute [tex]\( x = 2 \sec(\theta) \)[/tex] into the expression:
[tex]\[
25 x^2 - 100
\][/tex]
This becomes:
[tex]\[
25 (2 \sec(\theta))^2 - 100
\][/tex]
Next, calculate [tex]\( (2 \sec(\theta))^2 \)[/tex]:
[tex]\[
(2 \sec(\theta))^2 = 4 \sec^2(\theta)
\][/tex]
So, the expression transforms into:
[tex]\[
25 \cdot 4 \sec^2(\theta) - 100
\][/tex]
Simplify the terms inside the parenthesis:
[tex]\[
100 \sec^2(\theta) - 100
\][/tex]
Factor out the common factor of 100:
[tex]\[
100 (\sec^2(\theta) - 1)
\][/tex]
Now, recall the Pythagorean identity for secant:
[tex]\[
\sec^2(\theta) = 1 + \tan^2(\theta)
\][/tex]
So:
[tex]\[
\sec^2(\theta) - 1 = \tan^2(\theta)
\][/tex]
Thus, the expression simplifies to:
[tex]\[
100 (\tan^2(\theta))
\][/tex]
Now, take the square root of the entire expression:
[tex]\[
\sqrt{100 \tan^2(\theta)}
\][/tex]
This can be further simplified by taking the square root of each term:
[tex]\[
\sqrt{100} \cdot \sqrt{\tan^2(\theta)}
\][/tex]
[tex]\[
10 \cdot |\tan(\theta)|
\][/tex]
Since [tex]\(\tan(\theta)\)[/tex] is positive in the interval [tex]\( 0^\circ < \theta < 90^\circ \)[/tex], we can drop the absolute value:
[tex]\[
10 \tan(\theta)
\][/tex]
Thus, the simplified expression is:
[tex]\[
\boxed{10 \tan(\theta)}
\][/tex]
