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

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.

ParametricPlot can plot region:

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


• Thanks. It worked. Commented Feb 7 at 6:44
• @youthdoo glad was helpful. Enjoy time with Wolfram Language/Mathematica. :) Commented Feb 7 at 6:57

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}]}]


• Thanks too. I wonder why the edges seem to be a bit rough... Commented Feb 7 at 10:09
• @youthdoo RegionPlot` automatic discrete the region, it is not easy to get the exact boundary.
– GMT
Commented Feb 7 at 11:36