To tackle the given equation [tex]\(\frac{\sec^2 \theta - 1}{\sec^2 \theta} = \sin^2 \theta\)[/tex], let's go through a detailed, step-by-step solution.
1. Recall the Trigonometric Identities:
- [tex]\(\sec(\theta) = \frac{1}{\cos(\theta)}\)[/tex]
- [tex]\(\sec^2(\theta) = \frac{1}{\cos^2(\theta)}\)[/tex]
- [tex]\(\sin^2(\theta) + \cos^2(\theta) = 1\)[/tex]
- [tex]\(\tan^2(\theta) = \frac{\sin^2(\theta)}{\cos^2(\theta)}\)[/tex]
- Also, from the Pythagorean identity: [tex]\(1 + \tan^2(\theta) = \sec^2(\theta)\)[/tex]
2. Express the Left Side in Terms of Sine and Cosine:
The given equation is:
[tex]\[
\frac{\sec^2 \theta - 1}{\sec^2 \theta}
\][/tex]
Substitute [tex]\(\sec^2(\theta) = \frac{1}{\cos^2(\theta)}\)[/tex] into the equation:
[tex]\[
\frac{\frac{1}{\cos^2 \theta} - 1}{\frac{1}{\cos^2 \theta}}
\][/tex]
3. Simplify the Expression:
[tex]\[
\frac{\frac{1 - \cos^2 \theta}{\cos^2 \theta}}{\frac{1}{\cos^2 \theta}}
\][/tex]
Multiply numerator and denominator by [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[
\frac{1 - \cos^2 \theta}{\cos^2 \theta} \times \frac{\cos^2 \theta}{1}
\][/tex]
The [tex]\(\cos^2 \theta\)[/tex] in the numerator and denominator cancels out:
[tex]\[
\frac{1 - \cos^2 \theta}{1} = 1 - \cos^2 \theta
\][/tex]
4. Use the Pythagorean Identity:
We know that [tex]\(1 - \cos^2 \theta = \sin^2 \theta\)[/tex]:
[tex]\[
1 - \cos^2 \theta = \sin^2 \theta
\][/tex]
5. Conclude:
Therefore, we have shown that:
[tex]\[
\frac{\sec^2 \theta - 1}{\sec^2 \theta} = \sin^2 \theta
\][/tex]
Thus, through these steps, we've derived that the equation simplifies to:
[tex]\[
\sin^2(\theta) = \sin^2(\theta)
\][/tex]
