$\sin^n (x)$ is it same as $(sin \: x)^n$ , n is a Natural number [closed]

Yes, $\sin^n(x) = (\sin x)^n$. And yes, it does apply to all trig functions.

One thing to beware of, though: this does not apply to general function notation $f(x)$. Generally $f^2(x) \ne (f(x))^2$, instead $$f^2(x) = f \circ f(x) = f(f(x)) $$ Also $f^3(x) \ne (f(x))^3$, instead $$f^3(x)=f \circ f \circ f(x)=f(f(f(x))) $$ This notational pattern continues to higher iterates $f^n(x)$.

The short answer is Yes, the notations are the same. And you're right, the notation is unclear at times. That applies to the other trig functions too.

One prominent and confusing exception is the use of "arc" trig functions, which are the inverse of trig functions. Often they are denoted as $\sin^{-1}(x) = \arcsin(x)$ which is NOT the same as the reciprocal of the sine function $(\sin(x))^{-1} = \csc(x).$ Same goes for other trig functions, you gotta be careful with whether you mean the inverse or the reciprocal. But as long as you use copious parentheses it should be clear what your exponent indicates.

Note that this is not the same as an iterated trig function.