Um... not actually. The value of tan(15) is the same as the value of tan(45 - 30), but not at all the same as the value of tan(45) - tan(15)!
Are you familiar with
function notation at all? If so, then note that the tangent function is a
function, so "tan(x)" means "f(x)", where the specific name of the function "f" is "tangent". Just as f(a + b) does
not mean the same thing as f(a) + f(b), so also tan(45 - 30) does
not mean tan(45) - tan(30).
To simplify tan(45 - 30), you have to use the angle-difference formula they gave you.