What is cos²(x)?

Question

This looks odd to me. I need a definition. Is it just the square of $\cos(x) $ ?

Like $\ \cos^2(x) = \cos(x) \cdot \cos(x) $ ?

Then why don't you write it like that: $\cos(x)^2 $ ?

Answer 1

Yes, $\cos^2 (x)$ usually means $\cos(x) \cdot \cos(x)$.

Most other information already given here is also correct:

• $\cos^2 x$ is probably most common as shortest
• $(\cos(x))^2$ is most clear for beginners, but not practical - it has too much brackets, that are annoying to write and obscure equations.
• $\cos(x)^2$ can be understood as $\cos x^2 = \cos(x^2)$
• $\cos^2 x$ or $\cos^2 (x)$ can also mean $\cos(\cos(x))$. If you want to use this notation, you should note it, because it is less common. However, $\cos^{-1} x$ is often used instead of $\arccos (x)$, so often does not mean the same as $(\cos x)^{-1} = \frac{1}{\cos x} = \sec x$.

Sometimes additional brackets mean something, so it is not always safe to add them. For example $f^{(n)}$ denotes $n$th derivative of function $f$, like $f^{(2)}=f''$. You will learn what to use in practice. If you are not sure, you can always explain what do you mean. Writing that you use $\cos^n x = (\cos x)^n$ should be enough if it is not automatically clear that you are not using this notation for iterated function.

Answer 2

It is because the square is somehow "applied" on the function.

For function, you will probably often encounter this notation, and that is because, it is the function itself that is squared, not just the value.

For instance, $$\ln^2 : t \mapsto \ln(t)^2$$ Is a function, so if you write $\ln^2(t)$ it refeers to the value of the function $\ln^2$ evaluated at $t$, while if you write $\ln(t)^2$ you refeer to the value of the function $\ln$ evaluated at $t$ that you then square.