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[ Wiktionary : ]  [...]  Latin cottīdiānus, quōtīdiānus ‎(“happening every day”), from adverb cottīdiē, quōtīdiē ‎(“every day, daily”),
from an unattested adjective derived from quot ‎(“how many”) + locative form of diēs ‎(“day”).

[ Etymonline : ]  [...]  from Latin quotidianus "daily," from Latin quotus "how many? which in order or number?" (see quote (v.)) + dies "day" (see diurnal).  [...]

'quot' and 'quotus' appear a wrong, discordant choice of adjective to Compound dies, because they are used to connect or question a quantity with another noun'; but the frequency and quantity in 'every day' is evident: it is 'every'. So an adjective like omnis appears more reasonable.

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It's not 'every' day, but 'any' day. Think of it along the lines of 'how soever many days', i.e., all days.

See Cicero's Letters to Atticus 5.7 (or on Perseus) for a distinction between cotidie and in dies singulos. In the translation linked to above, we read:

Daily, or rather more and more every day, I send you shorter letters...

for:

cotidie vel potius in dies singulos breviores litteras ad te mitto...

This is a loose translation, but the distinction seems to be along the lines of:

On whatsoever day [you might pick/name], or, rather, on every single day, I send you shorter letters...

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