Hi Pauline,
I generally only use only the following  when caculating the relevant derivatives or integrals.

sin^2+cos^2 =1
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
sin(a+b)= sin(a)cos(b) + cos(a)sin(b)
d(sin(x))/ dx = cos(x)
d(cos(x))/ dx = - sin(x)

You can derive the rest of the formulas from the above. However as you practice more and more you tend to remember other formulas e.g. for tan, cot without much effort. At the beginning it takes some effort but with time it becomes easier.