Find range of cot^-1x+sec^-1x+cosec-1x....answer is coming (π/2,π/4] union [5π/4,3π/2)

Let f(x) = cot⁻¹x + sec⁻¹x + cosec⁻¹x. We need to determine its range.

Step 1: Define the Domains of Inverse Functions
The domain of sec⁻¹x is x ∈ (-∞, -1] ∪ [1, ∞).
The domain of cosec⁻¹x is x ∈ (-∞, -1] ∪ [1, ∞).
The domain of cot⁻¹x is x ∈ (-∞, ∞).
Since sec⁻¹x and cosec⁻¹x are only defined for x ≤ -1 or x ≥ 1, we restrict our function to these values.

Step 2: Use Standard Identities
Using inverse trigonometric identities:

cot⁻¹x + tan⁻¹x = π/2
Using x = secθ and x = cosecθ transformations,

cot⁻¹(secθ) = π/2 - tan⁻¹(secθ)

cosec⁻¹x = sin⁻¹(1/x)

sec⁻¹x = cos⁻¹(1/x)

Thus, we get a simplified form:

f(x) = cot⁻¹x + sec⁻¹x + cosec⁻¹x.

By considering various cases:

For x ≥ 1, the expression simplifies to values in the range (π/2, π/4].
For x ≤ -1, the expression simplifies to values in the range [5π/4, 3π/2).
Thus, the final range of f(x) is:

(π/2, π/4] ∪ [5π/4, 3π/2).