Question

What do you get when `[tan(x)]^2`? Is it `tan^2(x)` or `tan(x)^2`? Also is `tan(tan(x))` same as `tan^2(x)`?

Answer 1

`(tan(x))^2 = tan^2 x`

Explanation:

Expressions like `sin^2 x`, `cos^2 x` and `tan^2 x` are really shorthand for `(sin(x))^2`, `(cos(x))^2` and `(tan(x))^2` respectively.

Note that if conventions are not clear, then when we write `tan x^2` we could intend `tan(x^2)` or `(tan(x))^2`.

So the popular practice is to write `tan^2 x` when we mean `(tan(x))^2` and `tan(x^2)` when we mean `tan(x^2)`.

It certainly saves on parentheses, but gets a little confusing when you meet `tan^(-1) x`, which is not `(tan x)^(-1)` but the inverse tangent.

Note that `tan(tan(x))` is not the same as `tan^2(x)`.