I guess this is more of a math problem in a cryptography context so I apologize beforehand if it is not the right place to ask. Basically I have to check whether a certain implementation of RSA key-pair generation adheres to FIPS 186-4. More specifically, Appendix B-3-1.
FIPS 186-4 necessitates that $d$ (the private exponent) be created like so:

$d = (e^{-1})\bmod(\text{LCM}(p-1, \space q-1))$

The library in question(openssl v1.0.1) calculates $d$ like so:

$d = (e^{-1})\bmod((p-1)(q-1))$

I can't prove or disprove whether these two create the same set of answers for $d$.

The condition for generation of $p$ and $q$ is that $(p-1)$ and $(q-1)$ are both relatively prime to $e$ (the public exponent) so both formulas have answers.

Also since $p$ and $q$ are both prime, $(p-1)$ and $(q-1)$ will both be even numbers and from $a \times b=\text{GDC}(a, \space b) \times \text{LCM}(a, \space b)$ we know that $\text{GCD}(p-1, \space q-1) \geq 2$ so $\text{LCM}(p-1, \space q-1) \neq (p-1)(q-1)$.

My question is are they the same or different?

I would also appreciate if you could point me in the right direction math-wise so that I could potentially solve this myself.

P.S.: I understand for openssl v1, there is a FIPS module and also that openssl v3.0 will try to apply for a FIPS 140-2 certificate. I am unfortunately stuck with the version I mentioned and I cannot change that(it's not up to me).