Sure, let's go through the derivation of the expression [tex]\(\frac{\sec^2 \theta - 1}{\tan \alpha}\)[/tex] step-by-step.
1. Recognize Trigonometric Identities:
- We know that [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex].
- We also know that [tex]\(\sec^2 \theta\)[/tex] can be expressed using the Pythagorean identity: [tex]\(\sec^2 \theta = 1 + \tan^2 \theta\)[/tex].
2. Substituting the Identity:
- The expression [tex]\(\sec^2 \theta - 1\)[/tex] can be rewritten using the above identity:
[tex]\[
\sec^2 \theta - 1 = (1 + \tan^2 \theta) - 1 = \tan^2 \theta
\][/tex]
3. Simplify the Given Expression:
- Now substitute [tex]\(\sec^2 \theta - 1\)[/tex] with [tex]\(\tan^2 \theta\)[/tex]:
[tex]\[
\frac{\sec^2 \theta - 1}{\tan \alpha} = \frac{\tan^2 \theta}{\tan \alpha}
\][/tex]
- Notice that [tex]\(\theta\)[/tex] and [tex]\(\alpha\)[/tex] should reference the same angle for proper trigonometric simplification; assuming [tex]\(\theta = \alpha\)[/tex], we get:
[tex]\[
\frac{\tan^2 \theta}{\tan \theta}
\][/tex]
4. Simplify the Fraction:
- Simplify [tex]\(\frac{\tan^2 \theta}{\tan \theta}\)[/tex]:
[tex]\[
\frac{\tan^2 \theta}{\tan \theta} = \tan \theta
\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{\sec^2 \theta - 1}{\tan \alpha}\)[/tex] when the angles are consistent ([tex]\(\alpha = \theta\)[/tex]) is [tex]\(\tan \theta\)[/tex].
