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The problem asks to prove that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) given certain conditions in a right-angled triangle.

### Given:
- \( \sin \theta = \frac{p}{h} \), where \( p \) is the perpendicular (height) and \( h \) is the hypotenuse.
- \( \cos \theta = \frac{b}{h} \), where \( b \) is the base and \( h \) is the hypotenuse.
- We are asked to prove \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

### Proof:

1. **Start with the definition of tangent:**
   \[
   \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{p}{b}
   \]
   where \( p \) is the perpendicular (opposite side to \( \theta \)) and \( b \) is the base (adjacent side to \( \theta \)).

2. **Use the given expressions for \( \sin \theta \) and \( \cos \theta \):**
   - \( \sin \theta = \frac{p}{h} \)
   - \( \cos \theta = \frac{b}{h} \)

3. **Now compute \( \frac{\sin \theta}{\cos \theta} \):**
   \[
   \frac{\sin \theta}{\cos \theta} = \frac{\frac{p}{h}}{\frac{b}{h}} = \frac{p}{b}
   \]

4. **Conclusion:**
   Since \( \frac{p}{b} = \tan \theta \), we have proven that:
   \[
   \tan \theta = \frac{\sin \theta}{\cos \theta}
   \]

This completes the proof.

Let me know if you'd like further details or have any questions!

### Related Questions:
1. How do you derive the formula for \( \sin \theta \) in terms of opposite and hypotenuse in a right-angled triangle?
2. Can you explain the geometric meaning of \( \tan \theta \) in terms of the triangle’s sides?
3. What is the significance of the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) in trigonometry?
4. How can the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) be used in this context?
5. What are some other trigonometric identities involving \( \tan \theta \)?

### Tip:
Remember, the basic trigonometric identities like \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \sin^2 \theta + \cos^2 \theta = 1 \) form the foundation for more advanced trigonometric manipulations and equations.

In a right-angled triangle, the lengths of the perpendicular (height) and the base are p and b respectively, and the hypotenuse is h. Given that sin(θ) = p/h and cos(θ) = b/h, prove that tan(θ) = sin(θ) / cos(θ).

Learn how to prove that tan(θ) = sin(θ) / cos(θ) in a right-angled triangle. The problem explores key trigonometric identities involving sine, cosine, and tangent functions, suitable for students in Grades 9-12.

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