Related to this question.
This problem is the root reason why I was never good at math - I am horrible at all forms of simplification in mathematics, hence I am bad at proofs.
The problem is that there is no way I can "work through" a simplification problem - I either know the steps in advance or I can't do it at all. There is no "working it out", no fixed technique, no guarantees - that the technique you used for the first problem works in simplifying and proving an identity in the second problem. The best example of such problems are proving trig identities but there are numerous other examples.
A lot of people say practice is the answer, but I just end up hitting a wall and looking at the solution which does not add to my skill. The solution tells me nothing about how a person might arrive to the solution - it likes magic. There is only one and only one way to solve the problem and you have to find it among all other infinite wrong ways. There is nothing in the problem to "work out" - either you know it or you don't.
So is there a logical/deterministic way one can do and get better at simplification?
I am good at understanding logic, but I can't see patterns. Hence I am a good coder since that mostly involves breaking problem into sub-parts and combining the solutions.
Yes, there is! Many people these days are down on memorization of basic facts, because, quite rightly, they recognize that high-level thinking and problem-solving are more than rote memorization. However, if you ask a mathematician what $9\times 7$ is, they will immediately, without thinking about it, tell you that it's $63$. This is across the board for experts in general. One trait characteristic of experts, and this is well-documented in the book Why Don't Students Like School, by Daniel T. Willingham (highly recommended!), is that the expert is a master of the basic facts of the discipline.
The bottom line is this: being good at math is more than rote memory, but it includes memorizing the basic facts solidly. Once you've got those down, you don't have to use brain resources simply recalling those whenever you need them, you can use your brain resources on the higher-level thinking skills.