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If you have a single amplitude wave then a delayed playback could be inversion of the wave. Not very sophisticated, but good enough for canceling a single frequency.

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The original post stated unequivocally that noise cancellation works by playing a delayed version of the sound before deciding it was only going to cancel a single frequency.

So either the text has been re-ordered or OP is under the mistaken impression this would also work when dealing with a mix of frequencies.

All I know is that decent noise cancelling over-ear headphones (Bose QC35 in my case) are really good at cancelling exactly that sort of noise - the hum plus whoosh of a box fan. Given that they are ineffective for high frequencies but very effective for the low frequency part of the spectrum where most of the fan noise (or airplane roar, or bus engine) is, I also always assumed that they played the least-possible-delayed anti-wave to cancel out the wave. Doesn't work for speech, sadly, since that has enough high-frequency energy to be intelligible even through these phones.

OP is mistaken.

There is no discernable difference between a phase-shifted tone, and a delayed tone, except for the initial period where the tone begins.

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