[College Calculus] Derivatives of trigonometric functions

find f'(x)

f(x)=[sin(x)sec(x)]/[1+xtan(x)]

not sure if there is a faster way to solve this but I started by using the quotient rule and found that I needed to find the derivative of (sin(x)sec(x)). To do this do I have to use the product rule?

Also I need the derivative of (1+xtan(x)); do I also need to use the product rule with x*tan(x)?

Up to this point, I have assumed that I do, but when I apply it, I end up with half a page(exaggeration) of crap and I can't simplify further.