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Cost of Debt is:

(i) return (Annual return in %) to Debt investor which he will get from the company. (Investor point of view)

(ii) expense (Annual expense in %) payable by the company to investor. (companies point of view).

Under both, cost of debt is same. All the three formula mention by you is derived on the basis of that concept only.

Example-1: To understand return to investor:

Suppose a person invested100 today and get10 at one year end and also get back100 principle at one year end. what will be the Annual return in %?

Return = Annual Earning/Investment  = [Rs 10 + (100-100)]/100 = 10%

Example-2 To understand return to investor:

Suppose a person invested100 today and get rs 10 at one year end, Rs 10 at 2nd year end and also get back110 principle at 2nd year end then what will be the Annual return in %?

Return = Annual Earning/ Investment = [Rs 10 + (110-100)/2]/100 = 15%.

There was no derivation of 2nd formula. only concept is there. Your 2nd formula is based on Example-2.

Why we devided 110-100 by 2?  because we get value appreciation of10 in 2 year then what will be the appreciation for 1 year, it is5. As we are calculating annual reuturn we have to take only one year earnings.

But remember that we ignored time value of money in above example. That's why the cost of debt calulated in 2nd formula is approximate and not exact formula. This formula is applicable upto IPCC level only in Final that formula is not acceptable.

It is not possible to explain all the concept here.