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According to chapter 17 of Ross's Corporate Finance (Brazilian translation of 2nd edition),

$$ r_{WACC} = \frac{S}{S+B}r_S + \frac{B}{S+B}r_B(1 - T) $$

and

$$ r_S = r_0 + \frac{B}{S}(1 - T)(r_0 - r_B) $$

where $S$ is equity, $B$ is debt, $T$ is tax rate, $r_0$ is the unlevered cost of equity, $r_S$ is the levered cost of equity, and $r_B$ is the cost of debt.

By replacing $r_S$ in the first formula and simplifying, I get

$$ r_{WACC} = \Bigg{(}\frac{S + B(1 - T)}{S + B}\Bigg{)}r_0 $$

which would mean the weighted average cost of capital does not depend on the cost of debt, $r_B$. This formula yields the same WACC as the one in the book, and I checked on some other examples as well.

Did I get this right? If so, is this because the higher tax shield compensates the additional risk from higher interest payments? If not, what am I doing wrong?

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    $\begingroup$ The first equation is definitional and is always true. The second equation holds in the "pure Modigliani Miller case", where there is no "cost of financial distress". In this case the third equation also holds: the company pays less taxes to the government (the tax shield) and hence $WACC < r_0$ (also $r_s>r_0$ because of the higher risk to stockholders, and $r_d<r_0$ by assumption. The only loser is the government). Note however that if debt is very high the "cost of financial distress" cannot be neglected. $\endgroup$ – noob2 Apr 12 at 13:41
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As noted in the comments, you arrive at the correct conclusion, given your assumptions.

This result is usually referred to as the Modigliani-Miller theorem:

The basic theorem states that in the absence of taxes, bankruptcy costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is unaffected by how that firm is financed.

Since the value of the firm depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt, the Modigliani–Miller theorem is often called the capital structure irrelevance principle.

In practice adding leverage to a firm has a number of benefits in good economic conditions (due to tax benefits, agency cost reduction, juicing up returns for equity holders) but is detrimental in bad economic conditions (bankruptcy costs, inability to access cheap capital in a market with asymmetrical information).

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In response to ‘WACC may not depend on debt’. The weighted average (WACC) accounts for the minimum required return for a set capital structure. If a company has 100% debt financing that would affect the WACC. Similarly it can be seen that up till a point, increasing the level of debt reduces the WACC and more equity increases the WACC. The reason is that by increasing debt financing creates a tax deduction where more debt = less tax which is a cost advantage... that is until the interest expense on that tax rises higher than the tax deduction. As far as I know the formula is a basic weighted average WACC = Rp X weightRP + RE X weightRP + before tax cost of D X weight of D (1-tax rate)... The cost of debt is important to account for tax. If it was merely the cost of equity that was used when debt financing was used in the capital structure, the WACC would not account for the benefits of debt financing and the average cost would be higher

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How are you deriving $r_s$? If you use a WACC calculator like this one WACC calculator, the cost of equity definitely matters. Based on your derivation, let's look at an extreme case where a company is financed 100% through debt. It would say that the cost of debt is irrelevant, but that doesn't make sense.

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