Solving the system $\cos^2x+\sin^2y=1$, $\cos y\sin y=\cos x\sin x$

Question

I have a set of trigonometric equations as follows:

$$\cos^2(x)+\sin^2(y)=1$$
$$\cos(y)\sin(y)=\cos(x)\sin(x)$$

I have tried to plot these two graphs on desmos and it seems that two functions agree on the line $x=y+n\pi$. However, I don't see any clue in getting this relation and I am hard stuck right now.

Could anyone give me some hint on this? Thanks!

Answer 1

We have that

$$\cos^2x+\sin^2y=1 \iff \cos^2x=\cos^2 y \iff y=\pm x+2k\pi \, \lor\, y=\pi \pm x+2k\pi$$

See also the related

• Solving $\cos(3x) = \cos(2x)$