1. **State the problem:** Given $\csc \theta = \frac{5}{4}$ and $0^\circ < \theta < 90^\circ$, find the value of $\sec \theta$. 2. **Recall definitions:** - $\csc \theta = \frac{1}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ 3. **Find $\sin \theta$:** Since $\csc \theta = \frac{5}{4}$, then $$\sin \theta = \frac{1}{\csc \theta} = \frac{1}{\frac{5}{4}} = \frac{4}{5}.$$ 4. **Use Pythagorean identity:** $$\sin^2 \theta + \cos^2 \theta = 1.$$ Substitute $\sin \theta = \frac{4}{5}$: $$\left(\frac{4}{5}\right)^2 + \cos^2 \theta = 1$$ $$\frac{16}{25} + \cos^2 \theta = 1$$ $$\cos^2 \theta = 1 - \frac{16}{25} = \frac{9}{25}.$$ 5. **Find $\cos \theta$:** Since $0^\circ < \theta < 90^\circ$, $\cos \theta > 0$, so $$\cos \theta = \frac{3}{5}.$$ 6. **Find $\sec \theta$:** $$\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3}.$$ **Final answer:** $\boxed{\frac{5}{3}}$