The problem involves a right-angled triangle $XYZ$, with $XY = 5 \, \text{cm}$ and $XZ = 13 \, \text{cm}$. The goal is to find the values for the expressions given:

1. $\text{A: } \tan X + \tan Z$
2. $\text{B: } \cos X \cos Z - \sin X \sin Z$
3. $\text{C: } \sin X \cos Z + \cos X \sin Z$

Step 1: Identify triangle sides

Since this is a right-angled triangle, we can use the Pythagorean theorem to find the length of the third side $YZ$:

$YZ = \sqrt{XZ^2 - XY^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \, \text{cm}.$

Step 2: Calculate the trigonometric values

• For $\angle X$: $\sin X = \frac{XY}{XZ} = \frac{5}{13}, \quad \cos X = \frac{YZ}{XZ} = \frac{12}{13}.$

• For $\angle Z$ (since $Z$ is complementary to $X$, $\sin Z = \cos X$ and $\cos Z = \sin X$): $\sin Z = \frac{12}{13}, \quad \cos Z = \frac{5}{13}.$

Step 3: Evaluate the given expressions

1. A: $\tan X + \tan Z$ $\tan X = \frac{\sin X}{\cos X} = \frac{5}{12}, \quad \tan Z = \frac{\sin Z}{\cos Z} = \frac{12}{5}.$ So, $\tan X + \tan Z = \frac{5}{12} + \frac{12}{5} = \frac{25 + 144}{60} = \frac{169}{60}.$

2. B: $\cos X \cos Z - \sin X \sin Z$ (this is the formula for $\cos(X + Z)$, and since $X + Z = 90^\circ$, $\cos 90^\circ = 0$): [ \cos X \cos Z - \sin X \sin Z = 0.