<s>[INST] If $\sin x + \sin y = 1$ and $\cos x + \cos y = 0$, solve for $x$ and $y$ 
*

*$\sin x + \sin y = 1$

*$\cos x + \cos y = 0$


Any valid pair of $(x, y)$ is fine, as the restrictions on the board in the image below are obscured.
I got the question from chapter 26 of a comic called Yamada-kun.

How can I solve this equation?
 [/INST] There are identities for sum of sin and cos:
$$\sin(x)+\sin(y) = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)_.$$
$$\cos(x) + \cos(y) = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right).$$
Using the first equation tells us that $\cos\left(\frac{x-y}{2}\right)\neq 0.$ Therefore by the second equation, $\cos\left(\frac{x+y}{2}\right) = 0.$
This may or may not be useful in finishing the problem.
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