There is a confusing ambiguity when it concerns superscript notation for sin, cos and tan functions. For instance tan^2(x) currently means tan(x)*tan(x), whereas tan^-1(x) means arctan(x), aka the inverse tangent function, rather than 1/tan(x) aka the cotangent function.

To reduce ambiguity and confusion, the rules of math notation could be changed as follows:

• tan^2(x) would mean tan(tan(x)), since the exponent is a modifier on the function itself
• tan(x)^2 would mean tan(x)*tan(x), since the exponent is a modifier on the result of the function

That way it would also be consistent with the inverse tan function: 

• tan^-1(x) would be the inverse tan function aka arctan(x)
• tan(x)^-1 would be 1/tan(x) aka cot(x).

And the nice thing about this is that if you actually use the superscript for these values rather than the ^ notation, the parenthesis would still be redundant for the tan^n(x) case, however tan(x)^n would still need parenthesis, to prevent confusion with tan(x^n). You could still use the (tan x)^n of course.

This will also reduce weird behavior in programs like Wolfram Alpha. Right now it evaluates tan^-1(x) to arctan(x), but it evaluates tan^-2(x) to 1/(tan x)^2. This inconsistency would not exist if the above rules were followed.

That is why I am asking for a change regarding the mathematical notation of exponents and inverse functions for sin, cos, tan and their siblings.