arcsin(x)+arccos(x)

essentially, yes.

There is no difference between “arcsin” and “Arcsin”. Similarly for arccos.

I’ll first clarify a condition which needs to be true for an inverse function to exist.

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The domain is the set of values the function maps from (essentially the set of input values), and the codomain the set of values it maps to (the output values).

A “function”, strictly speaking, must be able to take any value in its domain and map it onto a single value in its codomain. Call this condition “*”, as I’ll refer to it later.

Suppose f is a function with domain A and codomain B.

(img1)

(In the first case, think how the function f(x)=x^2 maps has both f(2)=4 and f(-2)=4.)

Suppose f is a function with domain A and codomain B (as a above).

Then (assuming f^-1 exists), f^-1 will have domain B and codomain A. Think “reversing the arrows” in the picture above.

For f^-1 to exist, each value in B needs to be mapped onto a unique value in A. So the diagram would need to look something like the following:

(img2)


Both f and f^-1 need to be functions so they both need to satisfy “*”.

One can “improve” a function so that it satisfies “*” by restricting the domain and essentially excluding problematic values.

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Bringing this all back to the question at hand...

arcsin and arccos are the inverse functions of the functions sin and cos respectively.

But if we take sin and cos as one would normally think of them...

(img3)


then they don’t — strictly speaking — have inverses. For example, you’d have arcsin(0) = 0 and arcsin(0) = pi and arcsin(0) = 2pi and ...

And that doesn’t make any sense because 0 =/= pi =/= 2pi =/= ... . (It wouldn’t satisfy “*” above.)

Instead we need to restrict their domains, and the usual way to do this is:

(img4)


So, for sin, we restrict the domain to
-pi/2 <= x <= pi/2,
and for cos, we restrict it to
0 <= x <= pi.

Note. There’s no particular reason why we choose these ranges of values for x, it’s just done this way by convention.

Therefore we have
-pi/2 <= arcsin(sin(x)) <= pi/2
and
0 <= arccos(cos(x)) <= pi.

So if we use “sin(x)=cos(x - pi/2)” then sin(x) won’t be the “improved” function we’ve created. Instead it would be:

(img5)

(classic graph transformations)

Using “sin(x) = cos(pi/2 - x)” works a treat:

(img6)

(Again, think about graph transformations.)


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Footnote 1. The conventional use of “sin” and “cos” don’t use these restricted domains. Typing sin(3pi/2) into a calculator gives -1, surprising nobody. But whatever value of x (between -1 and 1) you type in, arcsin(x) will only ever give values between -pi/2 and pi/2.

Footnote 2. There’s technically a difference between the codomain and the range of a function. But here, you may assume that they have equivalent definitions.