why discount tax shield benefit at the cost of debt(pre-tax)?

We discount the tax shield at the pre-tax cost of debt (or if you want, at the risk free rate) because the tax shield carries the same risk as the debt interest on which it is based. (It is debatable whether we should discount at the return to debt investors or the risk free rate (in theory they would be the same, but in practice not), but the examiner allows either to get the marks).

When tax is payable does not affect the ungearing of the beta, only obviously the cash flows when you come to discount.

Have you watched my free lectures on both of these things? If not, then they may help you.

First, there are arguments for using either the pre-tax cost of debt, or the risk free rate (which is pre-tax). Either is allowed in the exam (even though obviously the answer ends up different).

The purpose is to discount the tax shield at a rate applicable to the level of risk associated with the tax benefit.

The argument for using the pre-tax cost of debt, is that the risk associated with the tax shield will be the same as the risk associated with the debt interest itself, and the risk of the debt interest is that incurred by the investors and so we use the investors required rate of return (which is the same as the pre-tax cost of debt).

The argument for using the risk-free rate (which is always pre-tax) is that in theory the debt interest is fixed and therefore risk-free, and that therefore the tax benefit on the interest will also be risk free. (The interest is fixed, but it is only risk-free if we ignore the risk of bankruptcy, which is a M&M assumption. In practice there is risk which is why the investors required return (the pre-tax cost of debt) is normally higher than the risk free rate.)

I do explain this in my free lectures.

That is correct 🙂