To simplify the expression [tex]\(\sec^2 \theta - 1\)[/tex], we can use a well-known trigonometric identity. Here’s a step-by-step guide:
1. Recall the Pythagorean Identity for Trigonometric Functions:
The Pythagorean identity states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
2. Express [tex]\(\sec \theta\)[/tex] in Terms of Cosine:
The secant function is the reciprocal of the cosine function:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
Therefore,
[tex]\[
\sec^2 \theta = \left( \frac{1}{\cos \theta} \right)^2 = \frac{1}{\cos^2 \theta}
\][/tex]
3. Use the Pythagorean Identity Again:
By rearranging the Pythagorean identity, we can express [tex]\(1 - \cos^2 \theta\)[/tex] in terms of [tex]\(\sin \theta\)[/tex]:
[tex]\[
1 = \sin^2 \theta + \cos^2 \theta \implies 1 - \cos^2 \theta = \sin^2 \theta
\][/tex]
4. Modify the Original Expression:
Now substitute [tex]\(\sec^2 \theta\)[/tex] with [tex]\(\frac{1}{\cos^2 \theta}\)[/tex] in the given expression:
[tex]\[
\sec^2 \theta - 1 = \frac{1}{\cos^2 \theta} - 1
\][/tex]
5. Combine the Terms Under a Common Denominator:
To combine these terms, write [tex]\(1\)[/tex] as [tex]\(\frac{\cos^2 \theta}{\cos^2 \theta}\)[/tex]:
[tex]\[
\frac{1}{\cos^2 \theta} - 1 = \frac{1}{\cos^2 \theta} - \frac{\cos^2 \theta}{\cos^2 \theta}
\][/tex]
Simplifying this, we get:
[tex]\[
\frac{1 - \cos^2 \theta}{\cos^2 \theta}
\][/tex]
6. Simplify the Numerator Using the Pythagorean Identity:
Recall that [tex]\(1 - \cos^2 \theta = \sin^2 \theta\)[/tex]. So substitute this into the expression:
[tex]\[
\frac{\sin^2 \theta}{\cos^2 \theta}
\][/tex]
7. Express the Simplified Form:
[tex]\(\frac{\sin^2 \theta}{\cos^2 \theta}\)[/tex] is simply [tex]\(\tan^2 \theta\)[/tex] by the definition of the tangent function:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} \implies \tan^2 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta}
\][/tex]
Therefore, the simplified form of [tex]\(\sec^2 \theta - 1\)[/tex] is:
[tex]\[
\tan^2 \theta
\][/tex]
