In a previous question I attempted to formalize the argument of going from one proof of an identity to another, which turned out to be harder than I thought. The thing is, while it may be impossible to rigorously define what I was trying to define, we can still have an intuitive idea of what I was trying to say but couldn't get across properly. So I'll ask this instead, as it doesn't rely on any formal definition of "trivial variants" as Zach Effman was saying in the comments.
The tl;dr version of the previous question is that there are a lot of nifty formulas for things like $\pi$. A lot of them, however, are independent of each other. For example, if we wanted to prove that $$ 2\frac{2}{\sqrt{2}}\frac{2}{\sqrt{2 + \sqrt{2}}}\frac{2}{\sqrt{2+\sqrt{2+\sqrt{2}}}}\cdots = \frac{2}{\operatorname{Li}_2(3-\sqrt{8})-\operatorname{Li}_2(\sqrt{8}-3)}\int_0^1{\frac{\arctan^2x}{\sqrt{1-x^2}}\text{d}x} $$ we could show that both sides are equal to $\pi$, and because $\pi = \pi$ the original identity is true. But is there a way of showing that it's true without using the fact that both sides are equal to $\pi$?
I'm not talking specifically about the above identity though. I'm just wondering, does anyone know of any clever proofs of identities like the one above, that don't require proving both sides are equal to the same constant independently?
I'm hoping this question will be a little bit easier to answer than the previous one because we can all develop an intuitive definition of what it means for a proof to be clever, and for a proof to not require showing both sides of an identity are equal to the same constant independently. For example, if you wanted to show that $A = c = B$ where $A$ and $B$ are the two independent identities, you could show that $A + 1 = c + 1$ and $B + 1 = c + 1$, and technically this would satisfy our non-rigorous definition of "independent proofs", but it certainly isn't clever by the human definition of clever.
