Plot region formed by all points $\left\{(\cos(x+y),\cos(x-y))\mid(x,y)\in\left(0,\frac\pi2\right)^2\right\}$

Question

How can I use Mathematica to plot the region $$\left\{\big(\cos(x+y),\cos(x-y)\big)\left\lvert(x,y)\in\left(0,\frac\pi2\right)^2\right.\right\}~?$$

I thought of using RegionPlot and "$\arccos$"s, but this wouldn't work if I were to plot the region $$\left\{\big(\cos\left(x^2+\sin y\right),\cos\left(\mathrm e^x-y\right)\big)\left\lvert(x,y)\in\left(0,\frac\pi2\right)^2\right.\right\}.$$ Another way might be to use two great loop with small increment of $x$, $y$, but that's not efficient.

Answer 1

ParametricPlot` can plot region:

ParametricPlot[{Cos[x + y], Cos[x - y]}, {x, 0, Pi/2}, {y, 0, Pi/2}]

Answer 2

Or use ParametricRegion + Region. (or RegionPlot)

reg = ParametricRegion[{Cos[x + y], 
    Cos[x - y]}, {{x, 0, Pi/2}, {y, 0, Pi/2}}];
Region[reg, BaseStyle -> {EdgeForm[Directive@{Thick, Pink}]}, 
 Axes -> True, Frame -> True]
reg//Area

1.

Clear[reg];
reg = ParametricRegion[{Cos[x^2 + Sin@y], 
    Cos[E^x - y]}, {{x, 0, Pi/2}, {y, 0, Pi/2}}];
Region[reg, 
 BaseStyle -> {EdgeForm[Directive@{AbsoluteThickness[3], Pink}]}]