how can I recover my private key from my passphrase and password I decided recently to get some BTC and bitcoin core was the storage solution I decided on. 

I downloaded the software, and all seemed good. No messages about storage or such. 

Around a month later I purchased BTC from an atm and sent it to the Core address. 

Upon returning home, I found it was "out of sync". I literally wiped everything off my 250gb Mac in order to accommodate it. Still nothing. 

I attempted to reduce the cache size, however Core told me I should reset, and so I did. However, this too didn't help.

I am trying to find how I can recover my private key from my passphrase and password.
I am trying to find how I can recover my private key from my passphrase and password


In general you cannot recover a private key from a wallet password or wallet passphrase.



If what you are calling a "passphrase" is really a "recovery phrase" or "seed phrase", then you can recover private keys from that. A recovery phrase typically consists of twelve or twenty-four words, usually generated for you by the wallet before use, usually from a specific word-list.

AFAIK current Bitcoin Core does not support recovery phrases.



If the wallet is still working, make a backup (File -&gt; Backup Wallet) or extract the wallet.dat file and recreate the wallet on a machine with more capacity -- or -- extract the private keys and import or sweep them into a wallet that supports SPV. Don't delete anything from your 250GB Mac until you have the new wallet working.