I don't get your point. What's the intention of the substitutions ?
Slightly off the point, but if you needed sides lengths for some pythagorean triangles, you could either trial and error your way with
or you could choose any m and n as given above to get valid results like this:
let m=7 and n=9 (or any other integers you like)
and you will see that
I am trying to find a similar way to break down
Then, all solutions of are precisely given by , , , where is any positive integer. Namely, any pair satsifying the above restrictions and any will produce a Pythagorean Triple.
Note that while what you wrote gives infinitely many solutions, not all solutions are produced this way (Example: the triple will not show, and still ).
It happens that the question on (*) has an answer depending on the above.
From equation (*), we see that are both odd or both even. So, the equation is equivalent to (simply verify this), where the fractions are actually integers.
The problem has reduced to another problem that we already know to solve.
There are two possibilities:
(1) , , . To find and explicitly we add and subtract and to get , .
(2) , , . In a similar way, , .
Example, I choose and apply part(1): Then . Indeed, .
Or, if we applied part(2) with : and it's true that .