1. **State the problem:** We need to find the cosecant of angle $Z$ in a right triangle with sides $YZ=68$, $XZ=32$, and $XY=60$, where the right angle is at $X$. 2. **Recall the definition:** Cosecant of an angle is the reciprocal of sine. That is, $$\csc(\theta) = \frac{1}{\sin(\theta)}.$$ For angle $Z$, $$\sin(Z) = \frac{\text{opposite side to } Z}{\text{hypotenuse}}.$$ 3. **Identify sides relative to angle $Z$:** The side opposite $Z$ is $XY=60$, and the hypotenuse is $YZ=68$. 4. **Calculate $\sin(Z)$:** $$\sin(Z) = \frac{60}{68}.$$ 5. **Simplify the fraction:** $$\frac{60}{68} = \frac{\cancel{4} \times 15}{\cancel{4} \times 17} = \frac{15}{17}.$$ 6. **Calculate $\csc(Z)$:** $$\csc(Z) = \frac{1}{\sin(Z)} = \frac{1}{\frac{15}{17}} = \frac{17}{15}.$$ 7. **Final answer:** $$\boxed{\frac{17}{15}}.$$