How to completely delete the head of a function expression

You can actually Delete the head of the expression, which is part 0:

Delete[#, 0] & /@ {Cos[a], Sin[b], Tan[c]}

{a, b, c}

With version 10 operator forms:

Delete[0] /@ {Cos[a], Sin[b], Tan[c]}

{a, b, c}

One case of interest may be held expressions. If our expression is:

expr = HoldComplete[2 + 2];

And the head we wish to remove is Plus, we cannot use these:

Identity @@@ expr
Sequence @@@ expr
expr /. Plus -> Identity
expr /. Plus -> Sequence
Replace[expr, _[x__] :> x, 1]

All produce e.g.:

HoldComplete[Identity[2, 2]]  (* or Sequence *)

We can use Delete or FlattenAt:

Delete[expr, {1, 0}]
FlattenAt[expr, 1]

HoldComplete[2, 2]
HoldComplete[2, 2]

You could also use a pattern that includes the surrounding expression on the right-hand-side, as demonstrated here, e.g.:

expr /. h_[_[x__]] :> h[x]

HoldComplete[2, 2]

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Notes

As the documentation for Delete reads:

Deleting the head of a whole expression makes the head be Sequence.

Delete[Cos[a], 0]

Sequence[a]

Since this resolves to a in normal evaluation this should usually not be an issue.

Does this come close?

Cos[a] /. Cos[a] -> a

Or

Cos[a] /. _[a] -> a

Or

First@Cos[a]

Or

list = {Sin@a, Cos@b};
First /@ list

{a, b}

You remove a head by replacing it with Identity

Cos[a] /. Cos -> Identity

For doing this over lots of expressions:

list = {ArcTan[x], ArcTan[x], ArcTan[x], Cot[x], Cot[z], Cot[x], 
  ArcTan[z], ArcTanh[y], ArcTanh[x], Cot[y]};

list[[All, 0]] = Identity

or

Identity @@@ list

etc