Orthogonal Matrix problems

This problem I have partially solved, but I'm having issues concerning format and proper proofs. The question is listed below.

Let and be orthogonal matrices. Determine which of the following are orthogonal (provide reasons for answers).

i) (my answer is below, but I'm not sure if it is concrete enough)

Let . Since is orthogonal, then . Therefore, is orthogonal.

ii)

(I'm afraid I'm not sure about this one)

If is also an orthogonal matrix, must be orthogonal?

I've inferred that is orthogonal, using the following three equivalent statements for some matrix :

a. is orthogonal.

b. for every

c. for every

The issue I'm having is SHOWING how C is orthogonal without 'begging the question' (without using circular logic).

Can anyone help me with this?