# How do you prove #arcsin x + arccos x = pi/2#?

##### 2 Answers

as shown

#### Explanation:

Let

then

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

#### Explanation:

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:

Show

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

First

Now

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,