Is it normal to train at the gym with your mouth taped shut?

I just joined a new gym this week, as a welcome gift I was given 4 free sessions with a personal trainer. After the initial assessment, e.g. BMI measurement, interests, goals, etc., She proposed a training programme designed to improve my cardio and help reduce muscle fatigue.

In my first session yesterday, she told me I had to do all of the exercises breathing only through my nose. This was supposed to make me breathe deeper and slower, thus improving the O2/CO2 exchange. It sounded like she really new what she was talking about, so I made the effort, but it was really difficult. I told her I didn't think I could manage it. She made do some tests to insure I was getting enough air through my nose. It turns out I was, but I found breathing through my mouth was a force of habit, and one she said I needed to break. Her solution was to put a piece of strapping tape over my mouth. She said she had used ithis method on loads of clients and it really works. I reluctantly agreed and trained the of the session with my mouth taped shut. I found it difficult to get the hang of at first, but after a while It actually felt easier, and I think my body kind of felt more energised? But I really felt embarrassed and thought everyone was secretly laughing at me. I felt I looked stupid with tape on my mouth, a bit silly and childish.

I am meant to do my second training session this week, and I am feeling a bit anxious and self conscious about it. Is it normal? Or should I just make the effort, stick to the training and try to not worry about what other people think?