If sum of the elements in the range of  f(x)=sin−1⁡x+cos−1⁡x+tan−1⁡x+tan−1⁡1x+sec−1⁡x+cosec−1⁡x is kπ, then k=

We know that (1) sin−1⁡x+cos−1⁡x=π2;∀x∈[−1,1] (2) tan−1⁡x+tan−1⁡1x=π2 if x≥0−π2 if x<0 (3) sec−1⁡(x)+cosec−1⁡(x)=π2;∀x∈(−∞,−1]∪[1,∞)Only -1 and 1 satisfies the given equationf(−1)=π2+−π2+π2=π2f(1)=π2+π2+π2=3π2∴ Range is π2,3π2 Sum of the elements in the range is 2π⇒k=2Therefore, the correct answer is (4).