> Is there a way to simplify expressions involving arctan and
> sin and cosine? For example:
> sin(((atan(2 * x * sqrt(x^2 + 1)))/2)) or
> cos(((atan(2 * x * sqrt(x^2 + 1)))/2))
There is no generic way to apply all known simplifications (though
presumably you could write one that does some sort of combinatorial
search).
The key insight in cases like this is to realize that you somehow want
to get the trig functions to apply directly to the inverse trig
functions. Instead of sin(f(atan(...))), you want f(sin(atan(...))).
So that suggests using the halfangle formula:
(C1) sin(((atan(2 * x * sqrt(x^2 + 1)))/2));
2
ATAN(2 x SQRT(x + 1))
(D1) SIN(----------------------)
2
(C2) %,halfangles;
1
SQRT(1 - -----------------------)
2 2
SQRT(4 x (x + 1) + 1)
(D2) ---------------------------------
SQRT(2)
(C3) radcan(%);
x
(D3) --------------
2
SQRT(2 x + 1)
To see what is going on here in "slow motion", try setting
triginverses:false before step C2.
Other approaches to try in general include using exponentialize and
logarc, followed by simplifying the resulting mess using
radcan/factor/etc., but that approach doesn't lead to a quick and easy
result in this case.
-s