Let q(x)=tanxcotx for all real x in the common domain of the two factors.
(a) Simplify q(x) to an equivalent elementary expression that involves no trigonometric products.
(b) Find q′(x) by differentiating your simplified expression from part (a).
(c) Without using the result of part (a), differentiate q(x)=tanxcotx directly and simplify your answer.
(d) Explain in complete sentences why the answers obtained in parts (b) and (c) must be identical, and discuss the advantages of algebraic simplification prior to differentiation in this context.