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.

Hello, zikcau25!

Welcome aboard!

I would start with the right side . . .


Multiply by

. .

. . . . . .


. . . . . .


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


We can start with the left side, too . . .



. . . . . .

You a Genius having dealt with it from beginning to end perfectly ... encountering no obstacle along the way through. Thank you very much.

But in my lengthy approach I confronted a second Puzzle reason for which I make this thread and which I previously simplify, to be analysed, in term of x (step 4. expressed as Algebraic Identity).

1. Here is my definite solution.

*UNCONVENTIONAL APPROACH:
- Most people would suggest to multiply each of the Binomials on the numerator (LHS) without a CLEAR JUSTIFICATION.
What matters is that it serves to resolve the problem.
What they don't know is that it is unconventional to general algebraic principles and lead to confusion.




*CONVENTIONAL SOLUTION & PROOF:



Now I came up with a solid explanation.

Hope you appreciate my work. Thank you.