ZDV Center Controller reported radio failure led to an inability to communicate and resulted in a loss of separation.

Narrative

Our radios are causing safety issues. This has been a known problem for years. The failure of our equipment caused this loss of separation and deep personal distress.Aircraft Y and Aircraft X were in conflict at FL360. Approximately 5 minutes before the loss of separation would occur; I issued Aircraft Y a descent clearance to FL340. There was no response and I issued the clearance three more times before coordinating backwards to earlier sectors to locate the aircraft. They did not have the aircraft on their frequencies. I did not have time to continue searching for Aircraft Y; so I issued a descent clearance to Aircraft X to FL350. There was no response. I issued the clearance again and still no response. My radios were working fine before the initial descent clearance to Aircraft Y; so I determined it was a headset issue and called over an On-the-Job Training Instructor (OJTI) in the area to plug into the sector and issue the clearance immediately. He transmitted several times with no response. We then determined that the radio must be the issue and I switched to Backup Emergency Communications (BUECs). He issued the clearance again and Aircraft X heard the clearance and descended. Separation was lost.We cannot provide safety services when our equipment doesn't work.Recommendation: Fix our radios. It has been a known issue for many years at ZDV. There is no excuse.