<s>[INST] Solving $2\cos(x) = \sin(x)$ How would you solve equations of the form $ a \sin (x+b) = \sin (x)$?
Eg. $ 2 \cos(x) = \sin(x) $
I realy have no idea how I would solve this kind of equations.
 [/INST] For your initial question, you can use the angle-sum rule for $\sin(\alpha + \beta):$ $$\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$$
So $$a \sin (x+b) = \sin (x) \iff a (\sin x \cos b + \cos x \sin b) = \sin x$$
We can divide through by $\cos x$, provided $\cos x \neq 0$, noting that $a, \cos b, \sin b$ are all constants.

For your example:
Note that if $\cos x \neq 0$, then $2 \cos x = \sin x \iff 2 = \dfrac {\sin x}{\cos x} = \tan x \iff \tan^{-1}(2) = x$.
And as @Alex notes,  $2\cos x = \sin x \implies \cos x \neq 0$.
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