I don't think that's as bad as it looks. Part of the point of inventing concise notation for mathematics is so that you can "let the formalism carry you throug

I don't think that's as bad as it looks. Part of the point of inventing concise notation for mathematics is so that you can "let the formalism carry you through"; a great deal of the time you can write down the next line of a proof by looking at the previous line or two and simplifying something without thinking too hard about what it means. One of the things that the students are practising is performing a longish series of operations while thinking about the important stuff and letting the simple stuff take care of itself. So I don't think it should be too concerning if a student starts to make mistakes like this: what's gone wrong may not be "the student doesn't understand what sin means", but "the student has gone astray in converting their understanding to an internalised rule of formal manipulation". (It's a bit like if a younger pupil mistakenly gets it stuck in their head that 6x7 is 49: it doesn't mean they've fundamentally misunderstood what multiplication is.)