Explanation

With u = sec(theta)+tan(theta) = p+sqrt(p^2+1) as in Q47, subtracting the conjugate equation gives 2tan(theta) = u-1/u = 2p, so tan(theta) = p. The conjugate is 1/u = sqrt(p^2+1)-p, from (p+sqrt(p^2+1))(sqrt(p^2+1)-p) = 1. Adding the equations isolates sec(theta) while subtracting isolates tan(theta) - mixing the two up is the common slip.