<s>[INST] Solve system of equations: $\sqrt2 \sin x = \sin y, \sqrt2\cos x = \sqrt5\cos y$ $\sqrt2 \sin x = \sin y, \sqrt2\cos x = \sqrt5\cos y$
I tried to use tangent half-angle substitution, tried to sum these two equations and get this
$\sin(x+ \pi/4) = \sqrt3 / \sqrt2 \sin(y+\tan^{-1}(\sqrt5))$
I stuck.
Any Help is appreciated!
 [/INST] Square both equations, then add them up. $2 \sin^2 x = \sin^2 y$ and $2 \cos^2 x = 5 \cos^2 y$, so $2 = \sin^2 y + 5 \cos^2 y$, and hence $1 = 4 \cos^2 y$. From here, $\cos y = \pm \frac1{2}$, and you should substitute back in to check if both work. You'll also get multiple values of $y$ from this, which you need to check with the $\sin$ relations.
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