Why is the area from the equator to latitude $x$ proportional to $\tan x$?

Question

When lengths are scaled by $\sec x$, area is scaled by _____. Why is the area from the equator to latitude $x$ proportional to $\tan x$?

This is a question about Mercator maps. I think the blank is $\int\sec x\,dx$. But I am not sure about the second question. $\int\sec x\,dx=\ln|\sec x+\tan x|+C$. There is a logarithm and a secant. How can it be proportional to $\tan x$?

Answer 1

When length is scaled by a factor $\alpha $, then the area is obviously scaled by a factor of $\alpha^2$. So here $\alpha =\sec x $.

The area from the equator to the latitude $x $ is then proportional to $\displaystyle \int \sec^2 x dx=\tan x $. Hope it helps.