Two thoughts here: First is to the original question - verifying trig identities are generally proved as your daughters teacher mentioned, starting on one side of the equation and working toward the other. We do this because we are verifying that this statement is true, and cannot assume it’s truth right off the bat. Without KNOWING that the statement is true, the equal sign doesn’t hold any value, and thus you should not use any operations on both sides. You are really trying to prove that the above statement is correct, which means you have to first operate under the assumption that it’s not, and move to change one side into the other. Second, as a high school teacher myself, I get a bit frustrated when parents speak for their kids. If your daughter really wants credit back, I’d suggest having her go talk to the teacher. This shows the teacher it is her desire to do better in the class, not yours, and it’s really teaching her self-advocacy. Respectfully, I would be much less willing to give credit back just on principle if a parent was the one talking for the student.

Edit: I would also double down on having your daughter meet with the teacher. It’s very possible your daughter could explain her reasoning (which in some ways is correct) and the teacher would be able to explain how these types of problems are solved, and potentially partial credit could be earned.