Solving tricky trigonometric inequalities can be challenging, but there are some helpful methods to make it easier. Here are some tips that I think work well:

1. Use One Trigonometric Function: Try to change the inequality to involve just one trigonometric function, like sine or cosine. This can make things simpler. For example, remember that $\tan(x) = \frac{\sin(x)}{\cos(x)}$, which can be useful.

2. Draw Graphs: Drawing the graphs of the functions can really help. By looking at the graphs, you can see where they cross each other or where one is above the other. This makes spotting the solution range easier.

3. Know Key Angles: Learning about key angles for sine, cosine, and tangent can show you where these functions are positive or negative. For example, knowing that $\sin(x) > 0$ in the first and second quadrants can help you find solutions faster.

4. Test Ranges: After finding where your functions are equal, check the ranges between those points. Just pick a number from each range and plug it back into the original inequality to see if it works.

5. Factor and Simplify: If you can break down the expressions into simpler parts, do it! This often makes solving the inequality much easier.

Using these methods, I have worked through some complicated trigonometric inequalities. Good luck!