You're correct; the answer is indeed 0. Let's re-examine the problem more carefully.

Corrected Solution:

1. Given:

   • Triangle $DEF$ is a right triangle with $\angle DFE = 90^\circ$.
   • $DF = 4$, $FE = 3$, and the hypotenuse $DE = 5$.
   • Point $G$ lies on $DE$.
   • The measure of $\angle DFG$ is $x$ and the measure of $\angle GFE$ is $y$.
   • We are asked to find the value of $\cos x - \sin y$.

2. Step 1: Analyze the triangle

   • In $\triangle DEF$, $\angle DFE = 90^\circ$, and the other two angles, $\angle FDE$ and $\angle DFE$, add up to $90^\circ$ as well.

3. Step 2: Express $x$ and $y$ in terms of known angles

   • Notice that $\angle DFG$ and $\angle GFE$ are complementary angles because $\angle DFG + \angle GFE = \angle DFE = 90^\circ$.
   • Therefore, $x = 90^\circ - y$.

4. Step 3: Substitute into the trigonometric expressions

   • We need to calculate $\cos x - \sin y$. Since $x = 90^\circ - y$, we have: $\cos(90^\circ - y) = \sin y$
     • Therefore: $\cos x - \sin y = \sin y - \sin y = 0$

Final Answer:

$\cos x - \sin y = 0$

Thank you for pointing that out! Do you have any more questions or need further clarification?

Here are some related questions:

1. What are complementary angles and how do they relate to trigonometric identities?
2. How can we use the sine and cosine of complementary angles to solve other problems?
3. What are other trigonometric identities involving angles?
4. How would this problem change if point $G$ were at a different location on $DE$?
5. What is the geometric interpretation of the result $\cos x - \sin y = 0$?

Tip: Complementary angles (those adding up to $90^\circ$) have trigonometric functions that swap, meaning $\cos(90^\circ - \theta) = \sin(\theta)$ and vice versa.