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There is a very simple explanation for your question which will clarify your thinking process:

Assume a company has $1mm in EBIT, is debt-free and has a tax rate of 40%, the company will have to pay $400K in taxes.

Now if the company has $1mm in interest to pay due to debt it has, then yes, the company will have to pay $1mm in interest, but it is only $600k greater than the $400k in taxes it has to pay anyways. Therefore the real cost of debt is really just the incremental $600k.

Simple?

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