Certainly! Let's delve into the given trigonometric identity step by step:
We want to prove the identity:
[tex]\[
\sec^2 \theta - \tan^2 \theta = 1
\][/tex]
### Step-by-Step Solution:
1. Recall the Definitions:
- The secant function is defined as [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
- The tangent function is defined as [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex].
2. Express [tex]\(\sec^2 \theta\)[/tex]:
- [tex]\(\sec^2 \theta = \left(\frac{1}{\cos \theta}\right)^2 = \frac{1}{\cos^2 \theta}\)[/tex].
3. Express [tex]\(\tan^2 \theta\)[/tex]:
- [tex]\(\tan^2 \theta = \left(\frac{\sin \theta}{\cos \theta}\right)^2 = \frac{\sin^2 \theta}{\cos^2 \theta}\)[/tex].
4. Substitute these into the left-hand side of the given identity:
- The left-hand side [tex]\(\sec^2 \theta - \tan^2 \theta\)[/tex] now becomes:
[tex]\[
\frac{1}{\cos^2 \theta} - \frac{\sin^2 \theta}{\cos^2 \theta}
\][/tex]
5. Combine the fractions:
- Since both terms have a common denominator [tex]\(\cos^2 \theta\)[/tex], we can combine them:
[tex]\[
\frac{1 - \sin^2 \theta}{\cos^2 \theta}
\][/tex]
6. Use the Pythagorean identity:
- Recall the Pythagorean identity: [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
- Rearranging this identity, we get: [tex]\(1 - \sin^2 \theta = \cos^2 \theta\)[/tex].
7. Substitute [tex]\(1 - \sin^2 \theta\)[/tex] with [tex]\(\cos^2 \theta\)[/tex]:
- So, [tex]\(\frac{1 - \sin^2 \theta}{\cos^2 \theta}\)[/tex] becomes:
[tex]\[
\frac{\cos^2 \theta}{\cos^2 \theta}
\][/tex]
8. Simplify the fraction:
- [tex]\(\frac{\cos^2 \theta}{\cos^2 \theta} = 1\)[/tex].
Therefore, we have shown that:
[tex]\[
\sec^2 \theta - \tan^2 \theta = 1
\][/tex]
So the identity is verified and the final result is indeed:
[tex]\[
1
\][/tex]
This concludes the step-by-step proof.
