Let q(x)=cosxsinx for all x where the expression is defined. The function q is known to be equal to tanx, but you will start from the quotient representation rather than the simpler form.
Respond to each part in order and write your work clearly, using complete mathematical sentences where justification is requested.
(a) Derive an explicit formula for q′(x) expressed entirely in terms of sinx and cosx without introducing any other trigonometric functions.
(b) Rewrite the expression obtained in part (a) so that no powers of sinx or cosx remain in the denominator. Simplify as far as possible.
(c) Explain why the final result in part (b) must agree with the well-known derivative of tanx on every interval where q is defined. Your explanation should reference the relationship between q and tanx as well as the properties of derivatives on domains consisting of multiple intervals.