Today we're pulling back the veil on the difference between the verbs show and reveal. To show something is to make it visible or known. The verb reveal is very close in meaning, but implies an element of discovery: items that are revealed have, up to the point of revelation, been invisible or concealed. Reveal comes from the Latin verb meaning "to unveil," which offers helpful imagery: when someone reveals something, they are in a sense removing a covering, whether literal or figurative, to show what lies beneath. Reveal is often found with stealthy adjectives, such as secret and hidden.

Something mutual is held in common or experienced between two parties, as in mutual friends. Similarly, reciprocal indicates a balanced relationship between two or more people, in which an act, thing, or feeling is given in return for another. Unlike mutual, reciprocal also carries the meaning of “matching or equivalent.” So a reciprocal promise may be made to ensure that both parties will maintain confidentiality about a matter. Often used in math, navigation, and grammar, reciprocal has a more technical implication of being one-for-one or the exact opposite or inverse of something, whereas mutual features in the world of finance or insurance when assets are shared (e.g., mutual funds). 

There are many types of knowledge, and erudition is one of them. Erudition is a thorough, formal, and profound sort of knowledge obtained by extensive research. The term is often used to discuss knowledge in fields other than those of mathematics and the physical sciences—so you're more likely to encounter it in discussion of philosophy or literature than in discussion of biology. Erudition is not a high-frequency word in English, but when it is used, it's likely to be in a context of glowing appreciation, as something that inspires respect or awe.