I've always struggled with these kinds of problems, since there's so many kinds of ways to do it. Often whenever i try to solve and prove identities I end up nowhere :/ So some tips and advice would be awesome.

The thing to understand is the difference between an identity and an equality .

For an equation in x, like you have

An identity has the same value or result, or is true

**for all x or for every x.**
An equality has the same value or result, or is true

**for ****some** x but **not **for every x.
So to prove that an equation is an identity you aim is to prove that it holds good for every value of x.

So

\(\displaystyle {x^2} - 4 = 0\)

is true for some values of x and is an equation

but

\(\displaystyle {\left( {x - 4} \right)^2} = {x^2} - 8x + 16\)

is true for all values of x so is an identity

Does this help?