Usually, $\sin^{2}(x)$ stands for $\sin(x)\sin(x)$, which is different from $\sin(x^{2})$ in general.

What was the original problem, and what were your original step-by-step attempt to this problem? There may be need to be more context for a good answer

I find no ambiguity and I agree with your professor. Your parenthesis in "$\sin(x)^2$" add no information compared to $\sin x^2,$ which means $\sin(x^2),$ while $(\sin x)^2$ is usually written $\sin^2x.$

@AnneBauval Of course it adds information. It says that $x$, and not $x^2$, is the argument of $\sin$. Would you say that $\Gamma(x)^2$ is equivalent to $\Gamma(x^2)$?

Sorry but $\sin(x)^2=\sin((x)^2)=\sin(x^2)=\sin x^2.$

@J.G. I mean, same applies. That's the square of the gamma function. I challenge you to find any reference in which that is used to mean $\Gamma((2x+1)^2)$.

@eyeballfrog I won't manage it, but there's a reason for that. My example differs from yours in two ways. Unlike $\Gamma$, $\sin$, the OP's function of interest, sometimes doesn't put brackets around its argument.

@MissMae The question gave f(x,y)=sin^7(4x-3y) and asked for ∂f/∂x and ∂f/∂y. I'm taking this class at community college, so the test was online and I only submitted the answer. My answer was ∂f/∂x= 28sin(4x-3y)^6*cos(4x-3y) and ∂f/∂y= -21sin(4x-3y)^6*cos(4x-3y)

@AnneBauval Wut. You're saying that rather than interpret the written parentheses as denoting the argument of $\sin$, I should assume they're meaningless and add in my own new parentheses, then say that's what the expression means.

@J.G. Indeed it doesn't always have brackets, but when it does have brackets I can't see how it's reasonable to ignore them and say it actually means something else.

@eyeballfrog I at least partially disagree. I think $\sin(2x+1)^2$ means $\sin y$ where $y=(2x+1)^2$, otherwise that can't be concisely communicated. Therefore, when I mean $\sin^2(2x+1)$ I say $\sin^2(2x+1)$. Granted, $\sin(x)^2$ might not be interpreted the same way.

Marking it as a duplicate. I disagree with your prof, but considering the spirited discussion above, I think it's best to avoid the ambiguity. Notation should clarify, not confound or stir unnecessary pedantic debates. This is a case of unfortunate notation.

Your teacher was wrong, sin²(x) = sin(x)² the parenthesis directly after the sine indicate that the function is what is squared and not the argument, I have never met a single person who wrote sin(x²) as sin(x)², maybe your teacher thought it was the case because he's the kind of person who doesn't use parenthesis around his sine

+1 for your teacher being wrong. Your comment above with the actual answer you posted is slightly different than writing $sin(x)^2$, but the same reasoning applies. The standard interpretation of $28sin(4x-3y)^6*cos(4x-3y)$ is $28sin^6(4x-3y)*cos(4x-3y)$. Although in maths there are cases where there are competing conventions, I don't think this is such a case - here, I would argue that a vast majority of maths people would interpret the preceding expression in the same way that you and I have vs. how your Professor did. Worth noting too, Wolfram interprets it in the way that you and I expect

But I also agree that it's best to avoid ambiguity and write $sin^6(4x-3y)$.

Sorry, but I'm confused. You're asking in the original post if $\sin^2(x) = \sin(x)^2$ (which IMO doesn't really matter as long as the context is clear), yet it seems like the motivation for that question comes from an online exam that asks for the partials of $f(x,y) = \sin^7(4x-3y)$. I also got the same answers as you after taking the partial derivatives. Did your professor mark you down for those answers or for writing "$(\sin(x))^2 = \sin(x)^2$"? I don't see how $\sin(x)$ comes into play here.

Anyone who would claim that $\sin(x)^2$ can be reasonably interpreted as $\sin(x^2)$ shouldn't be teaching math....

@Accelerator - I assumed the questioner used $x$ in place of $4x-3y$ to try to isolate the core issue.

Just to put the opposite argument, if I saw $\sin(4x-3y)^6$ in the writings of a competent mathematician, I would think they meant to compute the exponent first, because we already have the perfectly fine and widely understood notation $\sin^6$, and they chose not to use that. Plus, having the parentheses be around $4x-3y$, and not just $x$, makes it even more likely that they're there to make sure the exponent gets applied to the correct term. ....

I tell students that one of the things they are there to learn is how to communicate mathematical ideas clearly and correctly. Don't let the "your prof was wrong" crowd here influence you to charge in and demand full credit - you definitely should lose some points for writing something confusing. But my call would be that you deserve significant partial credit for nailing the key concept of partial derivatives.