Just as RDK said, there is no systematic way. If there was, everyone would be coasting through every question.
I mean, it doesn't get anymore systematic than
1. Be as familiar as possible (and on the look out for) identities that you know. The main ones being
sin2+cos2=1 and the two corresponding identities obtained by dividing this equation.
2. The whole goal with any problem with any new notation/function is getting rid of the new notation/function, which is usually achieved by isolating it. In the case of trig equations it's converting the equation into one in terms of just trig function.
Then it's easy to solve as normal. Keep that it mind and you should be fine.
And no it's not as black and white as you were thinking where you're supposed to decide what to convert to at the start - this isn't always possible. Just use the identities you know to simplify things and ultimately get the equation in terms of one trig function. I can guarantee the next time you get stuck, it will be because you either didn't apply one of the standard identities you know (purposefully of course) or you didn't make some sort of obvious simplification to the equation such as clearing fractions, expanding brackets etc. Or you just made a silly mistake lol.