NavList:

David Pike, you wrote:
"The two formulae are essentially the same, because secant(lat) = 1/cosine(lat).  However, the closer you get to the Pole, secant(lat) has a nasty habit of wandering off towards infinity."

Aha, that's interesting! I had not thought of that as a motivating factor here.

My guess --only a guess-- is something more mundane about calculation. The secant function was fundamentally a construct of the era of manual calculation. If you need to divide by the cosine of some angle, in logarithms that's equivalent to subtracting a logcos value, or it's equivalent to adding a logsec value. Adding is easier and less prone to error than subtraction. That made the logsec, and in other computations the logcsc, valuable on practical grounds. Now they are antiques, all but forgotten.

And when did this era of manual calculation (for a simple, short calculation, like this one) come to an end? Lots of options here, but I would say it was around 1975 when the first cheap "scientific calculators", like the TI-30, arrived on the market. I was still a "kid" in 1975, but I do remember pressing the cos and sin keys on those new calculators back then. The TI-30 took a while to come up with a value ...something like one or two seconds to display the sine of 45°! I had no idea what the numbers meant, but I was dying to learn...

Frank Reed