"Credulous" - Formal Use
Solution 1:
Credulous isn't a terribly common word, but it's not especially formal and can potentially be used in any register. The most important thing is its connotation: it means not just someone who is willing to believe, but implies a negative opinion of the person's intelligence and judgement. In this sense it's a near synonym of gullible.
Solution 2:
"Willing to believe" is a much more neutral-sounding phrase than "credulous", which is more along the lines of "willing to believe anything (no matter how far-fetched or improbable)".