SOLUTION: What is sin(arctan root3) + cos(arccot root3)?

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Question 669562: What is sin(arctan root3) + cos(arccot root3)?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
DISCLAIMER:
There is probably a cuter way to prove it,
invoking trigonometric identities and starting with
arctan%28sqrt%283%29%29=A<--->tan%28A%29=sqrt%283%29 and
arccot%28sqrt%283%29%29=B<--->cot%28B%29=sqrt%283%29.
However, my head hurts trying to think of it,
so I will go the way that is more natural to me.

The inverse trigonometric functions arctan%28x%29 and arccot%28x%29 are defined in a manner that makes sense.
We know that
sin%2830%5Eo%29=sin%28pi%2F6%29=1%2F2
cos%2830%5Eo%29=cos%28pi%2F6%29=sqrt%283%29%2F2 and

There are infinite other angles with cot%28x%29=sqrt%283%29,
but arccot%28sqrt%283%29%29=30%5Eo or pi%2F6
because it is defined as the angle between 0%5Eo and 180%5Eo (or between 0 and pi)
That has sqrt%283%29 for cotangent.
So cos%28arccot%28sqrt%283%29%29%29=cos%28pi%2F6%29=sqrt%283%29%2F2

We also know that
sin%2860%5Eo%29=sin%28pi%2F3%29=sqrt%283%29%2F2
cos%2860%5Eo%29=cos%28pi%2F3%29=1%2F2 and

and arctan%28sqrt%283%29%29=60%5Eo or pi%2F3
because arctan%28x%29 is defined with with a range of -pi%2F2 to pi%2F2 (-90%5Eo to 90%5Eo).
So sin%28arctan%28sqrt%283%29%29%29=sin%28pi%2F3%29=sqrt%283%29%2F2

In sum
Cute!