<s>[INST] Solve these equations simultaneously (trig) Solve for $ x,y: $ 
\begin{equation}\cos x -\cos(x+y) = 0
\end{equation}
\begin{equation}\cos y -\cos(x+y) = 0
\end{equation}
The answers are $(0, 0), (\frac{2\pi}{3}, \frac{2\pi}{3})$. 
I get $(0, 0)$, but how do you get the latter?
Sorry if this is a bit basic - I don't remember trig well...
tfa
 [/INST] Given
$$\cos x-\cos(x+y)=0\\
\cos y-\cos(x+y)=0$$
Subtracting the second equation from the first yields
$$\cos x=\cos y$$
Recall that $\cos x=\pm\sqrt{1-\sin^2x}$ and $\cos y=\pm\sqrt{1-\sin^2x}$.
This now becomes $\sqrt{1-\sin^2x}=\pm\sqrt{1-\sin^2y}$ yielding
$$\sin x=\pm\sin y$$
Recall that $\cos(x+y)=\cos x\cos y-\sin x\sin y$.
Thus, $\cos x-\cos(x+y)=0$ now becomes $\cos x-\cos^2x\pm\sin^2x=0$.
Solving $\cos x-\cos^2x-\sin^2x=0$ yields $\cos x=1$.
Solving $\cos x-\cos^2x+\sin^2x=0$ yields $\cos x-\cos^2x+1-\cos^2x=0$ which is equivalent to
$$(2\cos x+1)(-\cos x+1)=0$$
This yields two possibilities:  $\cos x=1$ or $\cos x=-\frac12$.  Can you do the rest?
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