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From my understanding, if something like this is given -> $f^2(x)$ then it is equal to $ff(x)$.

Now if that is the case, then shouldn't $\sin^2 (x)$ be equal to $\sin(\sin x)$, however, I have seen people use it as $\sin(x)^2$. Could someone please clarify my confusion.

Gerry Myerson
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Ved Rathi
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    $\sin^2 x =(\sin x)^2=(\sin x)\cdot (\sin x)$. For $\sin (\sin x)$ one possible notation is $\sin^{\circ 2} x$. – Gary Feb 22 '24 at 05:56
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    There's not much logic to it. $f^2(x)$ sometimes means $f(f(x))$, sometimes $(f(x))^2$. $\sin^2x$ always means $(\sin x)^2$, but $\sin^{-1}x$ usually means the functional inverse, not the multiplicative inverse. It's all conventions, and context. And it has probably been asked & answered on this site before. – Gerry Myerson Feb 22 '24 at 06:18

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This is a very annoying notational convention, but:

If $n$ is a positive integer, then $\sin^n(x)$ means $\big(\sin(x)\big)^n$. For instance, $\sin^2(x)$ means $\big(\sin(x)\big)^2$.

If $n$ is $-1$, then $\sin^{-1}(x)$ means the inverse function $\arcsin(x)$, instead of the reciprocal.

I'm not fully certain what happens for other values. I don't know if there's a consensus on what $“\sin^{-2}(x)”$ means.

The same stuff is true for the other trigonometric functions.

For what it's worth, for an arbitrary function like $f$, I like to write $f^{\circ n}(x)$ for $f(\dotsb(f(x)\dotsb)$, as described here, to avoid ambiguity like this. (In general, in mathematical writing, one should always explain such notations to the reader.)

Akiva Weinberger
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