First of all, the equation mentioned above works, however if you use Excel, you have to convert the angles to radians for the trig functions to work, then when done, you have to re-convert them back to degrees. Also, there is a "*" missing after the 3964.
HOWEVER, that equation does NOT give the values in the chart posted above, but instead it gives the values in the previous chart at the geo-orbit web page.
Ie the above equation gives declination numbers based on a satellite to your south, and if you use those, you'll be off by about 0.6 degrees on sats near the east or west horizon, because the declination of those sats is less by that amount because they are further away.
The chart posted above, which is referred to as "modified" declination, gives the MUCH better calculation which is based on the declination of sats near the east or west horizon, and what you should do to make it accurate for sats to the south is to add about 0.6 degrees to your latitude setting (or subtract from your elevation). You'll notice, that this is what they've done in what they have incorrectly called the elevation angle. It's actually the latitude plus about 0.6 in most cases, and the actual elevation to be used is 90 minus this number.
They don't give that calculation because it is a bit more complicated.
I have a calculator that does an approximation of what's done in the "modified" chart, at :
BJDISCALC2
I've made some assumptions for simplicity, so it isn't exact, but it's pretty close.
You can do a "page source" to see the equations I used.