Since this is multiple choice, we can pick any x from the listed intervals and check whether sec(x) < cot(x) is true.
• From 0 < x < π/2, take x = π/4. Then
sec(π/4) = 1/cos(π/4) = √2
cot(π/4) = 1/tan(π/4) = 1
but √2 < 1 is not true.
• From π/2 < x < π, take x = 3π/4. Then
sec(3π/4) = 1/cos(3π/4) = -√2
cot(3π/4) = 1/tan(3π/4) = -1
and -√2 < -1 is true. So sec(x) < cot(x) is always true for π/2 < x < π.
• From π < x < 3π/2, take x = 5π/4. Then
sec(5π/4) = 1/cos(5π/4) = -√2
cot(5π/4) = 1/tan(5π/4) = 1
and -√2 < 1 is true. So sec(x) < cot(x) is also true for π < x < 3π/2.
• From 3π/2 < x < 2π, take x = 7π/4. Then
sec(7π/4) = 1/cos(7π/4) = √2
cot(7π/4) = 1/tan(7π/4) = -1
but √2 < -1 is not true.
