How do you prove #arcsin x + arccos x = pi/2#?
The statement is true when the inverse trig functions refer to the principal values, but that requires more careful attention to show than the other answer provides.
When the inverse trig functions are considered multivalued, we get a more nuanced result, for example
We have to subtract to get
This one is trickier than it looks. The other answer doesn't pay it the proper respect.
A general convention is to use the small letter
The meaning of the sum of those is really every possible combination, and those wouldn't always give
Let's see how it works with the multivalued inverse trig functions first. Remember in general
We use our above general solution about the equality of cosines.
So we get the much more nebulous result,
(It's permissible to flip the sign on
Let's focus now on the principal values, which I write with capital letters:
The statement is indeed true for the principal values defined in the usual way.
The sum is only defined (until we get pretty deep into complex numbers) for
We'll look at each side of the equivalent
We'll take the cosine of both sides.
So without worrying about signs or principal values we're sure
The tricky part, the part that deserves respect, is the next step:
We have to tread carefully. Let's take the positive and negative
The principal value for the negative inverse cosine is the second quadrant,
So we have two angles in the second quadrant whose cosines are equal, and we can conclude the angles are equal. For
So either way,