How authentic is the Hadeeth that states that fasting the Day of ‘Arafah atones for the sins of two years and fasting the tenth of Muharram atones for the sins of one year?

All perfect praise be to Allaah, the Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

This Hadeeth is authentic and it was narrated by Muslim on the authority of Abu Qataadah, . The Hadeeth mentions that the Prophet, sallallaahu ‘alayhi wa sallam, was asked about observing fast on the Day of ‘Arafah and he, sallallaahu ‘alayhi wa sallam, said:  "It atones for the sins of the preceding year and the coming year."  Morevoer, he, sallallaahu ‘alayhi wa sallam, was asked about fasting the tenth of Muharram and he, sallallaahu ‘alayhi wa sallam, said: "It atones for the sins of the preceding year." This Hadeeth was also narrated by Ahmad and Abu Daawood on the authority of Abu Qataadah .

Thus, the Hadeeth that you mentioned is authentic and it is a Marfoo‘ Hadeeth (i.e., directly attributed to the Prophet, ).

Allaah Knows best.