To simplify the expression
[tex]\[
\frac{\sec^2(x) - 1}{\tan^2(x)},
\][/tex]
we can use some fundamental trigonometric identities. Here's a step-by-step breakdown:
1. Identify the trigonometric identity:
Recall the Pythagorean identity:
[tex]\[
\sec^2(x) = 1 + \tan^2(x)
\][/tex]
2. Substitute into the given expression:
We can now substitute [tex]\( \sec^2(x) = 1 + \tan^2(x) \)[/tex] into the expression:
[tex]\[
\sec^2(x) - 1 = (1 + \tan^2(x)) - 1
\][/tex]
3. Simplify the numerator:
[tex]\[
\sec^2(x) - 1 = \tan^2(x)
\][/tex]
4. Rewrite the original expression:
Substitute [tex]\( \tan^2(x) \)[/tex] for [tex]\( \sec^2(x) - 1 \)[/tex] in the numerator:
[tex]\[
\frac{\sec^2(x) - 1}{\tan^2(x)} = \frac{\tan^2(x)}{\tan^2(x)}
\][/tex]
5. Simplify the fraction:
Since the numerator and the denominator are the same, the fraction simplifies to 1:
[tex]\[
\frac{\tan^2(x)}{\tan^2(x)} = 1
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{1}
\][/tex]
