1. Why are there so many different formulae for trigonometry?

Because trigonometric functions do not obey the usual algebraic rules, separate rules have to be worked out for operations such as adding and multiplying trig. functions.

2. Do the trigonometric angle formulae have to be learnt off by heart?

Although it is handy and quicker to do problems if you know the formulae, they are given to you in the tables issued for most examinations.