Is $\sin^2(x)$ the same as $\sin x^2$? I'm working with derivatives and need to know if $\sin^2(x)$ the same as $\sin(x^2)$?
I almost don't want to ask because my last question was closed. It was a valid question and so is this one. I've been trying to find the answer on my own but the answers I've discovered are conflicting. Any clarification here is appreciated.
 A: No. $\sin^2(x)=(\sin(x))^2$. $\sin(x^2)$, on the other hand, is just $\sin(x^2)$.
A: No. $\sin(x^2)=\sin(x*x)$ while $\sin^2(x)=\sin(x)*\sin(x)$.
A: No, it's defined to be $sin^2(x)=(sin(x))^2$
A: Here is a graphical verification that these two functions are not equal. All the other answers are fine, but it is worth clarifying some other ambiguous notation in math.
Trig functions are notorious for the confusion they create when involving exponents. The conventions listed below are not necessarily optimal but being aware of them is helpful for the future:


*

*$\sin^2(x) = (\sin(x))^2$ "The square of the sine of $x$."

*$\sin(x^2)$ "The sine of $x^2$."

*$\sin^{-1}(x) = \arcsin(x)$ "The inverse sine of x. That is, if $y=\sin^{-1}(x)$, then $\sin(y) = x$."

*$\sin(x^{-1}) = \sin\left(\dfrac{1}{x}\right)$ "The sine of the reciprocal of $x$ (where $x\neq0$)."

*${(\sin(x))}^{-1} = \dfrac{1}{\sin(x)} = \csc(x)$ "The reciprocal of the sine of $x$. More commonly denoted the cosecant of $x$."


Other functions seem to adopt similar notation:


*

*$\ln^2(x) = (\ln(x))^2$ "The square of the natural logarithm of $x$."

*$\ln(x^2)$ "The natural logarithm of $x^2$."


But it only goes so far. I hope no one ever uses $-1$ as an exponent of the natural logarithm to indicate "the inverse natural logarithm of $x$"...because you would just denote that $e^x$. That is,


*

*$\ln^{-1}(x) = \dfrac{1}{\ln(x)} \neq e^x$ "The reciprocal of the natural logarithm of $x$ (where $x\neq 1$)."


The Golden Rule
Whatever notation you are using: if you have any hint of ambiguity, throw in another pair of parentheses in the appropriate places!
A: No, $\sin^2x =(\sin x)^2=(\sin x)\cdot (\sin x)$ and
$\sin x^2=\sin (x\cdot x)$ (in this case, the argument $x$ is in square).
A: No $\sin^2(x)$ means $(\sin x)^2$. This is not the same as $\sin(x^2)$.
