Given that \(\displaystyle 90o\le A\le 180^o\) means that \(\displaystyle 180^o\le 2A \le 360^o\) is in either the third or fourth quadrant.
Thank you, didn’t realise it was so simpleHello, and welcome to FMH!
Your working would be correct if the angle \(A\) were a first quadrant angle. But, it's a second quadrant angle. So, what must the sign of the cosine function there be?