@Carl Love I am working with matrices with integer entries, so I naturally made the assumption that they are all integers, however, this property does not require integers.

@Markiyan Hirnyk This is just what I needed. Thanks for a nice, simple solution!

@Carl Love It can be shown quite easily with such restrictions on the entries, the eigenvalues will be c+d and a-c. It's a nice little trick to have the eigenvalues be integer valued, without going for the obvious triangular approach.

@Kitonum Yes, I am aware of how to do it using the formulas for projections, etc. 

@Carl Love I wish I could upvote your comment!

@Axel Vogt I don' think learning math is the goal here ;-)

@Carl Love The polynomial in the question is x^3+x^2+2*x+1, but you used x^3+x^2+x+1.

@Carl Love I typically always work in 2-d mode, hence, I always use y'(x), p''(t), etc. :-)

@mehdi jafari I've tried the command with a trigonometric function in Maple 16, 17, and 18. It works fine in Maple 16, but fails in 17 and 18. So it appears to be a bug introduced in the newer versions of Maple. It would be nice to hear from a developer. :)

This is what I tested:

Student[Calculus1]:-Tangent(sin(3*x),x=0,output=plot,gridlines=true,caption="")

@mehdi jafari But yet, this works fine.

Student[Calculus1]:-Tangent(exp(-x^2),x=0,output=plot,gridlines=true,caption="")

I wonder why. Any ideas?

@Preben Alsholm Removing filled=true changes nothing. I do like the approach of using trace and multiple plots.

@Carl Love Any idea why for (about) Pi < theta < 5*Pi/4 and 7*Pi/4 < theta < 2*Pi there are regions shown that aren't part of the bounded region?

@Markiyan Hirnyk I apologize if my questions bore you. I did a search and the first few pages of results were not helpful because my search did not include the string "sum difference," or anything closely related. If you are indeed, bored by my queries, please ignore them. I do not think you need to be rude or demeaning because someone asks a question that you deem "unoriginal." Clearly, you have more experience with Maple, and I'm sure you're proud of that. That, however, does not give you the right to treat others with less experience with such disrespect. From what I understand MaplePrimes is designed to provide a forum where ALL users of Maple (and other products) can share and learn.

@Markiyan Hirnyk This solution is sufficient. I can confirm that it, indeed, works. I don't know if it is appropriate to pose another problem that it is closely related to this one, but here goes:

If I change the problem to 2*sin(2*x+Pi/4), and the desired form is sqrt(2)*sin(2*x)+sqrt(2)*cos(2*x), expand(,trig) no longer works for me. How might this one be tackled? Clearly, one may just replace "x" with "2x," but is this the only way?

@Carl Love I knew it had to be something simple. Your explanation is very good!