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The problem involves finding the value of \( \csc \theta \), given that \( \sec \beta = \frac{5}{4} \).

### Step-by-Step Solution:

1. **Recall the relationship between trigonometric ratios:**
   \[
   \sec \beta = \frac{1}{\cos \beta}
   \]
   Therefore, using the value given, we have:
   \[
   \sec \beta = \frac{5}{4} \quad \Rightarrow \quad \cos \beta = \frac{4}{5}
   \]

2. **Use the Pythagorean identity to find \( \sin \beta \):**
   \[
   \sin^2 \beta + \cos^2 \beta = 1
   \]
   Substituting \( \cos \beta = \frac{4}{5} \):
   \[
   \sin^2 \beta + \left( \frac{4}{5} \right)^2 = 1
   \]
   \[
   \sin^2 \beta + \frac{16}{25} = 1
   \]
   \[
   \sin^2 \beta = 1 - \frac{16}{25} = \frac{9}{25}
   \]
   \[
   \sin \beta = \frac{3}{5}
   \]

3. **Find \( \csc \beta \):**
   \[
   \csc \beta = \frac{1}{\sin \beta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}
   \]

### Conclusion:
The correct value of \( \csc \beta \) is \( \frac{5}{3} \), which corresponds to option **a**.

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Do you want further details, or have any questions?

Here are 5 related questions:
1. How can we find \( \tan \beta \) using the given values?
2. What is the relationship between \( \sec \beta \) and \( \sin \beta \)?
3. How do we derive the Pythagorean identity?
4. Can \( \csc \beta \) ever be a negative value?
5. What would \( \cot \beta \) be in this scenario?

**Tip**: Always check if your trigonometric values fit within the range of standard identities to avoid miscalculations.

Solve the trigonometric problem to find the value of cosecant csc θ using the relationship between secant and cosine. Includes step-by-step solutions for high school students studying Grades 9-12.

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