Q. If \( \sin x=\cos y \), what must be true about \( x \) and \( y \) ? answer choices \( x+y=90^{\circ} \) \( x=y \) \( x=2 y \) \( x+y=180^{\circ} \)

To solve the problem, we start with the given equation: \[ \sin x = \cos y \] We can use the co-function identity, which states that: \[ \sin x = \cos(90^\circ - x) \] This means that if \(\sin x = \cos y\), then we can set: \[ y = 90^\circ - x \] Now, if we rearrange this equation, we get: \[ x + y = 90^\circ \] Thus, the correct answer is: \[ \boxed{x + y = 90^\circ} \] Now, let's analyze the other options to confirm that they are not true: 1. **\( x = y \)**: This is not necessarily true since \( \sin x \) and \( \cos y \) can be equal without \( x \) and \( y \) being the same angle. 2. **\( x = 2y \)**: This does not follow from the given equation and is not a general truth. 3. **\( x + y = 180^\circ \)**: This is also not true based on the co-function identity we used. Therefore, the only statement that must be true about \( x \) and \( y \) is: \[ x + y = 90^\circ \]