WEIRD QUESTION HELP: ? Letting x = ANOTHER x ?

Question

When an identity or rule is written in x like:

sin(2x)=2cos(x)sin(x)

Then sin(2x^2)=2sin(x^2)cos(x^2) ...by the substitution x=x^2

but they're not the same x ....so it's a valid substitution?

And I'm making these substitutions that involve two xs ...without even thinking about it when I do algebra ...

My question is ...how do you think about it ..

do you think about this everytime youre given a definition youve never seen before ....or no ....is it a property that I should just know intuitively ...thats always works ...and not to think about it

becaue now that I'm thinking about it, its slowing me down ...I've done up to Calc III

THANKS

Answer 1

normally I see this written as another variable such as g=x^2 or f(g) rather than f(x). I think sooner or later this will bite you, so you should change your habit.