<s>[INST] trignometric equation, can i move the 2cos(2x) to the other part of the equation? When solving $2\cos(2x)+\sin(x)=1/2$ can i move the first $2$ from $\cos$ to the other parcel and multiply like this: $\cos(2x)+\sin(x)=1/4$?
 [/INST] No you may not.  If you'd like to divide by 2, you must divide both terms on the LHS by 2.  Not just your $2\cos(2x)$ term.  If you divided both terms by two, you'd be left with $\cos(2x)+\frac{1}{2}\sin(x)=\frac{1}{4}$  Try playing around with the double angle formula to solve this problem.
For example:
$2\cos(2x)=2(1-2\sin^2x)$,
So, we have $2-4\sin^2x+\sin x=\frac{1}{2}$.
Multiply both sides of the equation by 2 to get rid of fractions and you're left with $4-8\sin^2x+2\sin x=1$.  Let $u=\sin x$ and set equation equal to zero and you have $-8u^2+2u+3=0$.  Factor and we have $(2u+1)(-4u+3)=0$.  Which gives $u=-\frac{1}{2}$ and $u=\frac{3}{4}$.  Substituting back in we have $\sin x=-\frac{1}{2}$ and $\sin x=\frac{3}{4}$. $\sin x=\frac{1}{2}$ at $\frac{4\pi}{3}+2\pi k$ and $\frac{5\pi}{3}+2\pi k$. $\sin x=\frac{3}{4}$ at $x=\sin^{-1}(\frac{3}{4})$.  
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