For any y ∈ R, let cot⁻¹(y) ∈ (0, π) and tan⁻¹(y) ∈ (−π/2, π/2). Then the sum of all the solutions of the equation tan⁻¹(6y / 9 − y²) + cot⁻¹((9 − y²) / 6y) = 2π/3 for 0

Question

For any y ∈ R, let cot⁻¹(y) ∈ (0, π) and tan⁻¹(y) ∈ (−π/2, π/2). Then the sum of all the solutions of the equation tan⁻¹(6y / 9 − y²) + cot⁻¹((9 − y²) / 6y) = 2π/3 for 0 < |y| < 3, is equal to

Accepted Answer

4√3 − 6. The given equation involves inverse trigonometric functions and their complementary properties. Simplifying the expressions and solving for y within the given range, the sum of all solutions is found to be 4√3 − 6.

For any y ∈ R, let cot⁻¹(y) ∈ (0, π) and tan⁻¹(y) ∈ (−π/2, π/2). Then the sum of all the solutions of the equation tan⁻¹(6y / 9 − y²) + cot⁻¹((9 − y²) / 6y) = 2π/3 for 0 < |y| < 3, is equal to

Solution

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