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The problem provided is:

- \( \sin(x) = \frac{1}{3} \) 
- \( \sec(y) = \frac{5}{4} \), where \( x \) and \( y \) lie between \( 0 \) and \( \frac{\pi}{2} \).
- We need to evaluate the expression \( \cos(x + y) \).

### Step 1: Finding \( \cos(x) \)
Given \( \sin(x) = \frac{1}{3} \), we can use the Pythagorean identity:
\[
\sin^2(x) + \cos^2(x) = 1
\]
Substituting \( \sin(x) = \frac{1}{3} \):
\[
\left(\frac{1}{3}\right)^2 + \cos^2(x) = 1
\]
\[
\frac{1}{9} + \cos^2(x) = 1
\]
\[
\cos^2(x) = 1 - \frac{1}{9} = \frac{8}{9}
\]
\[
\cos(x) = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}
\]

### Step 2: Finding \( \cos(y) \)
Given \( \sec(y) = \frac{5}{4} \), we know that:
\[
\sec(y) = \frac{1}{\cos(y)} \Rightarrow \cos(y) = \frac{4}{5}
\]

### Step 3: Using the Angle Sum Identity for Cosine
The cosine of the sum of two angles is given by:
\[
\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)
\]
We already know \( \cos(x) \), \( \cos(y) \), and \( \sin(x) \). Now, we need to find \( \sin(y) \).

### Step 4: Finding \( \sin(y) \)
Using the Pythagorean identity again for \( y \):
\[
\sin^2(y) + \cos^2(y) = 1
\]
Substituting \( \cos(y) = \frac{4}{5} \):
\[
\sin^2(y) + \left(\frac{4}{5}\right)^2 = 1
\]
\[
\sin^2(y) + \frac{16}{25} = 1
\]
\[
\sin^2(y) = 1 - \frac{16}{25} = \frac{9}{25}
\]
\[
\sin(y) = \frac{3}{5}
\]

### Step 5: Final Calculation of \( \cos(x + y) \)
Now substitute all the known values into the angle sum identity:
\[
\cos(x + y) = \left(\frac{2\sqrt{2}}{3}\right)\left(\frac{4}{5}\right) - \left(\frac{1}{3}\right)\left(\frac{3}{5}\right)
\]
\[
\cos(x + y) = \frac{8\sqrt{2}}{15} - \frac{3}{15}
\]
\[
\cos(x + y) = \frac{8\sqrt{2} - 3}{15}
\]

Thus, the value of \( \cos(x + y) \) is \( \frac{8\sqrt{2} - 3}{15} \).

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Do you need further details or have any questions?

### Related Questions:
1. How do you derive the Pythagorean identity?
2. What are the steps for using the angle sum identities for sine and cosine?
3. How do you handle trigonometric problems when the angles are in different quadrants?
4. Can you explain the unit circle and its importance in trigonometry?
5. How do you simplify trigonometric expressions with radical terms?

### Tip:
Always double-check the quadrant where the angle lies to ensure correct sign usage in trigonometric functions.

If sin(x) = 1/3 and sec(y) = 5/4, where x and y lie between 0 and π/2, evaluate the expression cos(x + y).

Learn how to evaluate the expression cos(x + y) given sin(x) = 1/3 and sec(y) = 5/4, with x and y between 0 and π/2. This problem covers key trigonometric concepts, including the Pythagorean Identity and the Angle Sum Identity. Suitable for students in Grades 10-12.

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