$\"\"$

The calculation of a point on a Mercator projection contains the expression $\"\"$ , where $\"\"$ is the latitude of the point.

The expression is $\"\"$ .

Apply tangent sum identity : $\"\"$ .

$\"\"$

Apply half-angle identity : $\"\"$ .

$\"\"$

Thus, $\"\"$ .

$\"\"$

The expression is $\"\"$ .

Substitute $\"\"$ in above expression.

$\"\"$

a. $\"\"$ .

b. $\"\"$ .