# What is cos²(x)?

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: $\cos(x)^2$ ?

• Because $\cos{(x)^2}$ looks too much like $\cos{(x^2)}$ which would be incorrect. Jan 17, 2015 at 18:49
• Because it allows for removing brackets - the ultimate goal of mathematicians :-) --> $cos^2x$ Jan 17, 2015 at 18:50
• it is better to write $(\cos(x))^2$ Jan 17, 2015 at 18:59
• $\cos^2 x$ is even, not odd. ;) Jan 17, 2015 at 19:12
• Jan 7, 2017 at 22:42

It's the same as $[\cos(x)]^2$, which is really how this should be written. But it's kept around for historical reasons.

Truthfully, the notation $\cos^2(x)$ should actually mean $\cos(\cos(x)) = (\cos \circ \cos)(x)$, that is, the 2nd iteration or compositional power of $\cos$ with itself, because on an arbitrary space of self-functions on a given set, the natural "multiplication" operation is composition of those functions, and the power is applied to the function symbol itself, not the whole evaluation expression. But historically, the notation meant squaring the value of the trigonometric function. For an arbitrary function however, putting the power on the function symbol indicates compositional power. This historical notation though is also inconsistent, because $\cos^{-1}(x)$ does not usually mean $\frac{1}{\cos(x)}$ (i.e. $\sec(x)$) but rather $\arccos(x)$, thus a compositional power (iteration) now!

• Alas, mathematics is all plagued with inconsistencies. Dec 14, 2022 at 17:32

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.

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.

• Just to totally confuse things, $cos^{-1}(x)$ is not $sec(x)$ Jan 17, 2015 at 19:50