is it possible to solve this problem tan(arcsec(13/5)) and get an exact value without using a calculator?
Yes. Draw out a right triangle in the first quadrant. The hypotenuse will be 13 and the leg on the x-axis will be 5. (So the arcsecant of the angle will be 13/5!) Use the pythagorean theorem to solve for the vertical leg, which will be 12. Find the tangent of the angle, which will be 12/5.