1≠2

I recently recieved a math test back from my teacher during our Trig unit. I got an 89%, which is of course a very respectable grade. However, I couldn't believe it when I looked over the actual test, and realized that I lost 4 points, or 5% of the grade, over one stupid question. While this is not the actual question, it is basically what she was asking:

Prove the following identity:
2 sin x cos x - sin² x - cos² x = 1

For those who have not had Trig, I should tell you that one of the identities is that -sin² x - cos² x = -1. Therefore, the above equation could be rewritten as:

2 sin x cos x - 1 = 1

I then simplified the equation further, by adding one to both sides.

2 sin x cos x = 2

From there I went, divided by two, and proved the identity. My problem, according to my teacher, was in the adding of 1 to both sides of the equation. Because the two sides have not been proven to be equal, she said, I cannot add 1 to both sides.

Frankly, this is utter bullshit. If I have the equation 1=2, which is clearly not correct, adding 1 to both sides, yielding 2=3, is also not correct. There is no way that doing something to both sides of an inequality will make them an equality. No possible way.

I guess this is just my way of venting off some anger, but also asking, "What do you think? Should I get those 4 points back, and get that grade up to a 94%?"