Correct answer: (i) arcsec x < 2 or arcsec x > 4, i.e. x in [1, sec 2)... with the range restriction giving sec x in valid set; (ii) proof
(i) Put t = arcsec x in [0,pi/2) U (pi/2,pi]. (t-2)(t-4) > 0 means t < 2 or t > 4. Since the maximum of arcsec is pi (~3.14) < 4, the condition t > 4 is impossible. So we need arcsec x < 2 (and within the allowed range), which translates to a set of x values via the monotonic branches of sec. (ii) Given sin² x + sin² y < 1: sin² x < cos² y. Then tan² x * tan² y = (sin² x/cos² x)(sin² y/cos² y). Using sin² x < cos² y and sin² y < cos² x gives sin² x sin² y < cos² x cos² y, i.e. tan² x tan² y < 1, so |tan x tan y| < 1, ensuring arcsin(tan x tan y) is defined and strictly inside (-pi/2, pi/2).