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I am trying to learn the rules for the em dash. My colleagues and I were having an argument about this. One wrote this sentence for his students

We have just derived a fundamental trigonometric identity—the Pythagorean identity.

I suggested that the em dash cannot properly take the place of a comma. My colleague, however, insisted this was proper punctuation usage for emphasis. I have my doubts but I am not an expert. I would appreciate any input on this.

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    I would say that the em-dash is a better choice than a comma in the above text. (Though there should be white space on both sides of the dash.)
    – Hot Licks
    Aug 7, 2021 at 0:51
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    @HotLicks Many US style manuals (such as the Chicago Manual of Style) do not use spaces around the em-dash. The AP style manual is a notable exception. However, 'In British usage, an en dash (with space before and after) is usually preferred to the em dash as punctuation in running text, a practice that is followed by some non-British publications as well' (CMoS, 6.83).
    – linguisticturn
    Aug 7, 2021 at 4:12
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    @linguisticturn - A lot has to do with the typeface.
    – Hot Licks
    Aug 7, 2021 at 11:51
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    @HotLicks Ha, that's something I didn't know about! So you are saying that there are examples of style manuals that give different recommendations depending on the typeface used?
    – linguisticturn
    Aug 7, 2021 at 13:09
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    Does this answer your question? Usage of brackets/parentheses The terminal (precising) appositive the Pythagorean identity is a parenthetical, which may be offset by a comma, a colon (getting old-fashioned), a dash (most Brits use the spaced en-dash), brackets, or an ellipsis. Zero punctuation is not an option in this case. Different choices emphasise different pause-lengths, dramatic emphases.
    – Edwin Ashworth
    Aug 7, 2021 at 14:16

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Your colleague is correct.

Here is what the Chicago Manual of Style says about it:

6.85: Em dashes instead of commas, parentheses, or colons

The em dash, often simply called the dash, is the most commonly used and most versatile of the dashes. (In British usage, spaced en dashes are used in place of em dashes; see 6.83.) Em dashes are used to set off an amplifying or explanatory element and in that sense can function as an alternative to parentheses (second and third examples), commas (fourth and fifth examples), or a colon (first example)—especially when an abrupt break in thought is called for.

It was a revival of the most potent image in modern democracy—the revolutionary idea.
The influence of three impressionists—Monet, Sisley, and Degas—is obvious in her work.
The chancellor—he had been awake half the night—came down in an angry mood.
She outlined the strategy—a strategy that would, she hoped, secure the peace.
My friends—that is, my former friends—ganged up on me.

To avoid confusion, the em dash should never be used within or immediately following another element set off by an em dash (or pair of em dashes). Use parentheses or commas instead.

The Whipplesworth conference—which had already been interrupted by three demonstrations (the last bordering on violence)—was adjourned promptly.

or

The Whipplesworth conference—which had already been interrupted by three demonstrations, the last bordering on violence—was adjourned promptly.

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