Calculating area of raster pixel when pixels are different sizes at different latitudes using ArcGIS Desktop?

I have a large raster file of total mangrove cover for the whole globe (obtained from NASA: http://sedac.ciesin.columbia.edu/data/set/lulc-global-mangrove-forests-distribution-2000/metadata). I’m trying to calculate the area of mangrove forest that fall within protected areas (that are defined by polygon layer).

I can calculate the number of pixels that fall within the polygon, however, as the area of the pixels changes at differing latitudes, how can I reliably convert these pixel counts into area?

I’m aware that flattening a sphere causes some sort of distortion, so I tried Define Projection tool to change to World Cylindrical Equal Area (from WGS84), but that just shrinks the whole raster down to an unusable scale.

If I set the coordinate system of the layer to World Cylindrical Equal Area before importing the raster, it preserves the scale, but after measuring the pixels with a ruler, I can see they change at different latitudes.

I’ve included some screenshots for clarification.

The first image is the global mangrove forest cover (dark green), the second and third images are the >Layer Properties >Source window of the mangrove.tiff raster file (unaltered). And the last image is a measurement of 1 pixel close to the equator.

mangrove forest (dark green)
layer properties 1
layer properties 2
pixel

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