Why is $\cos(x)^2$ written as $\cos^2(x)$? I'm just wondering why the square of $\cos(x)$ (i.e.: $(\cos(x))*(\cos(x))$) is almost universally written in the form $\cos^2(x)$ rather than $\cos(x)^2$.  This seems particularly bizarre when one considers that $\cos^{-1}(x) \ne \cos(x)^{-1}$.
 A: Because we can safely drop the brackets without losing ambiguity, which means less effort when writing it out by hand.
$$
\cos^2x = \cos^2(x) = \cos(x)^2 \neq \cos x^2
$$
A: There are several competing notations. These seem to be the standard interpretations. The goal seems to be to use the least number of parenthesis and still be understandable. 
$\left .
\begin{matrix}
    \cos(\cos(x)) \\
    (\cos(x))^2 \\
\end{matrix}
\right\} = \cos^2(x) = \cos(x)^2$
$\left .
\begin{matrix}
    \dfrac{1}{cos(x)} \\
    \arccos(x)
\end{matrix}
\right\} = \cos^{-1}(x)$
$\cos(x^2) = \cos(x)^2 = \cos x^2$
Please note that $\cos(x)^2$ is the most ambiguous of the group and I personally feel that it should be avoided as much as possible.
Generally, the context should make it clear which meaning is being used.
A: This is because composition of functions are very rare when you are talking about trigonometric functions.
For any other $f: \mathbb{D} \to \mathbb{R}$, it may make sense to calculate $f(f(x))$, however for $\sin(x)$ or $\cos(x)$, composition like $\cos(\cos(x))$ is not a frequent use. That's why a misunderstanding in $\cos^2(x)$ is not so much in concern.
On the other hand, when it is about $\arcsin(x)$ and $\csc(x)$, there are conflicts about the use of $\sin^{-1}(x)$.
A: Well because when you write 
$$
\cos(x)^2 
$$
it can be misunderstood as cosine of (square of x) not the square of the whole value so to avoid confusion its written as
$$\cos^2(x)=(\cos(x))^2$$
