ImplicitRegion fails on apparently simple case

Consider the shaded region bounded by $sin x$, $cos x$, and $tan x$:

enter image description here

We can define this as an ImplicitRegion by:

(ScriptCapitalR) = 
 ImplicitRegion((0 < x < 1) (And) (y < Cos(x)) (And) 
     (y < Tan(x)) (And) (y > Sin(x)),
  {x, y})

However, RegionPlot((ScriptCapitalR)) fails to yield a figure after 15 minutes (v 11.3, MacOS).

However if I’m “smart” and put the bounds as $0<x< pi/4$ I do get a plot.

Moreover, its area,

RegionMeasure((ScriptCapitalR))

does not give an analytic solution (even after RootReduce, Simplify, etc.) even though an analytic form exists.

(One can get a numerical value through N@RegionMeasure((ScriptCapitalR)), but I seek the analytic solution.)

I’ve tried various forms based on RegionIntersection() and such, without success.

Of course I can use traditional calculus through Integrate and finding intersection points, but I’d like to compute the area more directly.

How can I 1) plot the region and (more importantly) 2) compute the analytic area?