<s>[INST] How to simplify the equation $ a\sin x\cos y + b\cos x\sin y = c\sin(2x)$? I am trying to derive the relation between x and y which are two physical parameters.I am trying to express x in terms of y, a and b. Is there a way to do it?
 [/INST] You can rewrite your equation as
$$b \cos x \sin y = \sin x \;( 2 \cos x - a \cos y ) $$
Squaring, we have
$$b^2 \cos^2 x \sin^2 y = \sin^2 x (2 \cos x - a \cos y )^2$$
So, writing $p:= \cos x$ and $q := \cos y$,
$$b^2 p^2 (1-q^2) = (1-p^2) (2p - a q )^2$$
This gives a quartic polynomial in $p$:
$$4 c^2 p^4 - 4 a c p^3 q + ( b^2  - 4 c^2 + a^2 q^2 - b^2 q^2 ) p^2 + 4 a c p q - a^2 q^2  = 0 $$
which does not have nice roots.
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