Let f(x)=sec−1⁡1+cos2⁡x, where  [.]  denotes the greatest integer function. Then the

f(x)=sec−1⁡1+cos2⁡xf(x) is defined if 1+cos2⁡x≤−1 or 1+cos2⁡x≥1 i.e., cos2⁡x≤−2 (not possible) or cos2⁡x≥0 i.e., cos2⁡x≥0 or x∈R Now, 0≤cos2⁡x≤1 or 1≤1+cos2⁡x≤2 or 1+cos2⁡x=1,2 or sec−1⁡1+cos2⁡x=sec−1⁡1,sec−1⁡2 Hence, the range is sec−1⁡1,sec−1⁡2