Hi, Mathsnewbie. I will try to point in the right direction on the first exercise of your post.
You may find that and are relevant. The table involving trig functions and their derivtives at Differentiation of trigonometric functions - Wikipedia, the free encyclopedia may also be of some use. I think this (along with the Fundamental Theorem of Calculus, i.e. taking antiderivatives) is enough to finish off the exercise.
If this is too cryptic let me know and I will try to be more explicit with my hints. Good luck!