Sure, let's break down the trigonometric identity [tex]\(\frac{\sec \theta}{\csc \theta} = \tan \theta\)[/tex] step by step:
1. Understand the Trigonometric Functions:
- [tex]\(\sec \theta\)[/tex] (secant of theta) is defined as [tex]\(\frac{1}{\cos \theta}\)[/tex].
- [tex]\(\csc \theta\)[/tex] (cosecant of theta) is defined as [tex]\(\frac{1}{\sin \theta}\)[/tex].
- [tex]\(\tan \theta\)[/tex] (tangent of theta) is defined as [tex]\(\frac{\sin \theta}{\cos \theta}\)[/tex].
2. Express [tex]\(\sec \theta\)[/tex] and [tex]\(\csc \theta\)[/tex] in terms of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
3. Substitute these expressions into the given equation:
Given: [tex]\(\frac{\sec \theta}{\csc \theta}\)[/tex]
Substitute [tex]\(\sec \theta\)[/tex] and [tex]\(\csc \theta\)[/tex]:
[tex]\[
\frac{\sec \theta}{\csc \theta} = \frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}}
\][/tex]
4. Simplify the Right-Hand Side:
Remember that dividing by a fraction is equivalent to multiplying by its reciprocal. Hence:
[tex]\[
\frac{\frac{1}{\cos \theta}}{\frac{1}{\sin \theta}} = \frac{1}{\cos \theta} \times \frac{\sin \theta}{1} = \frac{\sin \theta}{\cos \theta}
\][/tex]
5. Recognize the Simplified Form:
The simplified form [tex]\(\frac{\sin \theta}{\cos \theta}\)[/tex] is exactly how the tangent function ([tex]\(\tan \theta\)[/tex]) is defined:
[tex]\[
\frac{\sin \theta}{\cos \theta} = \tan \theta
\][/tex]
6. Conclusion:
After simplifying, we see that:
[tex]\[
\frac{\sec \theta}{\csc \theta} = \tan \theta
\][/tex]
Therefore, we have shown that [tex]\(\frac{\sec \theta}{\csc \theta} = \tan \theta\)[/tex] is indeed a valid trigonometric identity.
