The range of fx=sec-1log3tanx+logtanx3 is

fx=sec-1log3tanx+logtanx3.            =sec-1log3tanx+1log3tanxNow,  for log3tanx    to get defined,  tanx∈0,∞or   log3tanx∈-∞,∞  or  log3tanx∈RAlso,  x+1x≤-2  or  x+1x≥2i.e.,    log3tanx+1log3tanx≤-2or    log3tanx+1log3tanx≥2i.e.,    sec-1log3tanx+1log3tanx≤sec-1-2or     sec-1log3tanx+1log3tanx≥sec-12i.e.,     fx≤2π3  or  fx≥π3i.e.,fx∈π3,π2∪π2,2π3