They form a triangle...are you certain you've copied the problem exactly as stated?
What is the significance of the word "TEST" in the title of the thread? And why is it in all-caps?
Assuming the question is something similar to what you posted, you will need to find an equation for the tangent line. A point on this line is x=1 and y=-2tan(1); now all you need is the slope.
Yes, i am aware that it forms a triangle. However, my teacher said that we must use the equation that we get in y=mx+b form (after finding the derivative, etc) and rearrange it to find the rectangle area.
With the information given the only rectangle I can conceive of finding is one that sides lie on the x and y axis. Once you you know a point on the line (1,-2tan(1) ), and its slope = f'(1) = ? then you can plug those numbers into y=mx+b solve for b.. then solve for the y and x intercept which will give you the length and width of the rectangle.
Note: the above is making a lot of assumptions about the rectangle, that aren't "mathematically valid" from the original wording of the question. That is, there is no reasonable argument that this is THE rectangle you should be finding, except for the fact that given the information its the most reasonable one.