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The answer is 2 but I don't think your formulas are correct. I'm not even sure what you are thinking.
The area is simply A = (2 * sin(t)) * (2 *cos(t))
Take the derivative, equate to zero, and then isolate for the variable, t.
Now, isolating for a variable after taking the derivative does not always give you the maximum! It merely gives you a local extrema, which may or may not be what you are looking for. It just so happens that, in this question, the closest extrema to t = 0 is also the maximal area...
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