Why is it that cos(x) * cos(x) = cos^2(x) as opposed to cos(x)^2 ?
Is this something to do with the trig identities? What identities are useful to know off by heart?
It's a convention and, as will all conventions, it eliminates ambiguity. Nothing to do with identities, they still hold.
Is equal to (a lot of brackets, especially if you have lots of trig functions) or ?
IMO there are only four trig identities you need to know
You can derive all your other identities from these
Although it is not a good notation, many people don't write the ( ) in sin(x), just sin x. That would make it too easy to confuse (sin(x))^2 with sin(x^2). Of course, you can write (sin(x))^2 (with the parentheses) but it is simpler to write sin^2(x). In general, if f(x) is a function of x, f^2(x) means f(x)*f(x)= (f(x))^2.