Should I be afraid of using known identities that I can't prove?

Question

Recently I've noticed a pattern in all of my "researches" (if you can call them that), and that is I will not allow myself to use known identities if I can't prove them (or at least understand a given proof of them). For example, I was recently looking into integrals of Bessel Functions, and I read about the Sonine-Formula, that is, $$ \int_0^\infty{J_z(at)J_z(bt)J_z(ct)t^{1-z}dt} = 2^{z-1}\Delta(a, b, c)^{2z-1}\left(\sqrt{\pi}\Gamma\left(z+\frac{1}{2}\right)(abc)^z\right)^{-1} $$ where $\Delta(a, b, c)$ is the area of a triangle with side lengths $a, b$, and $c$ (or 0 if it doesn't exist).

This might be a bit of an extreme example, as I don't think I'd ever come across this at the level I'm working at, but if I did, I wouldn't go ahead and use it. Instead I'd be stopped dead in my tracks. I wouldn't allow myself to proceed until I knew exactly what was going on this formula.

I do this simply because I'm scared of what might happen if I don't fully understand the tools that I'm using.

But I've also noticed that a lot of higher level mathematics makes reference to identities not found or entirely understood by the original mathematician.

So should I be scared of not knowing everything about a problem? Should I readily use other peoples' work regardless of whether or not I understand it?

EDIT: It might be useful to note that I'm still only in high school, so I'm nowhere near the level of experience most of the people have on this website, and this fear I'm feeling might just be a side-effect of being young and inexperienced.

Answer 1

You'll never be able to know the proof of every mathematical truth. If you find a peer-reviewed result, you should feel comfortable using it. Some proofs span hundreds, even thousands, of very terse pages.

That said, if a particular identity is central to what you're researching you should try to learn a lot about it.

Answer 2

While I basically concur with David Peterson's answer one important thing to keep in mind is that such identities might pose restrictions, which may prove slightly too strict for your requirements. In that case, you should not blindly trust in the identity still being valid despite leaving its (currently established) boundaries.

As a simple example take the identity

$$ (-1)^{2n} = 1\ \forall n\in\mathbb Z$$

(which reads "even powers of minus one are plus one"). Now at some point you notice in your case $n$ might also take real values, in which case this "identity" actually becomes a falsehood and must be corrected to

$$ (-1)^{2r} = e^{2\pi i r},\quad |(-1)^{2r}| = 1\quad \forall r\in\mathbb R$$

(which reads "for real exponents, powers of minus one become phase factors, i.e. complex numbers of absolute value one"). And that also becomes invalid once you allow for complex valued exponents...