Type I and II errors accounted for by one equation?

On this website:

https://clincalc.com/stats/samplesize.aspx

both type I and II errors are accounted for by the one equation.

Surely, because a type I error involves too much positive variance and a type II error involves too much negative variance, the $$2$$ are mutually exclusive?

Will I not need to calculate the minimum sample size given a power of $$x$$ and a minimum sample size given an Alpha of $$y$$ and then choose the larger of the $$2$$ as one, the power say, is my limiting factor?

I appreciate both theoretical and practical feedback.

I'm looking for this in the context of the difference between 2 sample proportions if that makes a difference.

(In fact a longitudinal randomised control trial so 2x2)

• That's not the right interpretation of the power analysis. $\alpha$ is the significance level, not the type 1 error rate. If the null is true, $\alpha$ = type 1 error for tests of the correct size (which, for instance, Fisher's Exact Test is not). But by virtue of calculating power, the null is not true. – AdamO Jan 17 at 19:15