The second question first: Method of undetermined coefficients.
For the first, have you ever encountered the term "envelope?" (Sorry, couldn't find a good definition from the web.) An envelope is kind of a limiting solution. In this case it's a sine wave. As your solution mentions, the sin(delta*t) term has a very long wavelength compared to the other sine term. So sketch the sin(delta*t) term. Then we have a varying amplitude:
(or so since delta is so small.) This will be a "wave inside a wave." How can you tell? Either graph it using, say, Wolfram|Alpha or you have seen this kind of problem before.
-Dan