1. Why do we need to differentiate trigonometric functions?
One reason is that sometimes there is a need to find the gradient of a trigonometric graph.
2. Do I need to be able to prove all of the derivatives of the six basic trig functions from first principles?
There is probably no need to be able to prove them but you should certainly be able to follow and understand these proofs.
3. Do I need to memorise the derivatives of the six basic trigonometric functions?
You should try an memorise them even though they are provided in the Formulae and Tables Booklet provided with the Bursary examination paper.