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Studying the method of undetermined coefficients right now, at one point in an example problem I see this:

$$ (A-3B)\sin(t) + (B + 3A)\cos(t) = \sin(t) $$

What just happened?

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    the equation has to be true for any $t$? – 2017-02-12
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    That seems to be the implication. I'd include more details to clarify but this is all the worked example offers in terms of explanation. – 2017-02-12
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    I wrote what I think it is. Please take a look. – 2017-02-12
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    Ah, okay I get it. I thought I was missing a trig identity or something. Thanks. – 2017-02-12
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    you are very welcome – 2017-02-12

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If you want $A$ and $B$ such that the equation holds for any value of $t$ then take some particular values of $t$, such as:

$$t=0 \to B+3A=0\\ t=\frac{\pi}{2}\to A-3B=1$$

Now solve the system and get $A=\frac{1}{10}$ and $B=-\frac{3}{10}$