Fix for bug [ 1370433 ] trigsimp(sqrt(%i2)) != sqrt(trigsimp(%i2))?



A quick summary of the bug:

expr:2*(cos(x)^2-sin(x)^2)+2;

sqrt(trigsimp(expr)) -> 2*abs(x)
trigsimp(sqrt(expr)) -> -2*abs(cos(x))

I would expect these to be the same.

The issue is the call to radcan in trigsimp in
share/trigonometry/trgsmp.mac:

trigsimp(x):=
  trigsimp3(radcan(apply1(x,
                          trigrule1,trigrule2,trigrule3,trigrule4,
			  htrigrule1,htrigrule2,htrigrule3,htrigrule4)))$

The call to radcan is the source of this problem.  If radcan is
removed or replaced by ratsimp, the problem goes away.  Tracing radcan
shows what's happening: it simplifies sqrt(2*(cos(x)^2-sin(x)^2)+2) to
sqrt(2)*%i*sqrt(sin(x)^2-cos(x)^2-1).  trigsimp3 simplifies the sqrt
part to sqrt(-2*cos(x)^2) -> sqrt(2)*%i*abs(cos(x)).  Hence the
-2*abs(cos(x)). 

I think we should change this to be ratsimp instead of radcan.

Thoughts?

Ray