It's my first day on this forum.

I was working on a trigonometric identities question this evening, when I surprisingly got stuck.

It was only after that I discover a new simple unlearned principle in the question that I was able to finish it up.

Trigonometric identities most often need binomial expansion approach to it.

In my case, it requires a bit more - Sometimes unexpected challenge show up in the question - and it is exactly what I am going to share with you.

To some this may be a mild thing, but to me today it a great day today. lol

Firstly, I would like to discover the ways proposed by others to deal with the problem,

Before I may post my own solution I discover.

1. Question

2. Working ...

3. Simplifying ...

4. This was where I got stuck,

This is an exemplary case, where I can't find any known/taught (in textbook) clue to solve this problem, but with careful study I could find it.

... I would wish to see your solution to prove this "simple" Algebraic Identity.