Trigonometry & Inverse Trigonometry Question - None

Question

The value of x satisfying (cot⁻¹x)(tan⁻¹x) + 2($\frac{π}{2}$ − cot⁻¹x) − 3tan⁻¹x − 3($\frac{π}{2}$) ≥ 0 is

Accepted Answer

Step 1: Using the identity $ an^{-1}(x) + \cot^{-1}(x) = \frac{π}{2}$, we have $ an^{-1}(x) = \frac{π}{2} - \cot^{-1}(x)$

Step 2: Substituting u = cot⁻¹x:

(cot⁻¹x − 3)(2 − cot⁻¹x) ≥ 0

Step 3: Rearranging:

(cot⁻¹x − 3)(cot⁻¹x − 2) ≤ 0

Step 4: This inequality is satisfied when:

2 ≤ cot⁻¹x ≤ 3

Step 5: Since cot⁻¹ is a decreasing function:

cot 3 ≤ x ≤ cot 2

∴ Answer is (b) x ∈ (cot 3, cot 2)

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