Question
If tan⁻¹(x) + tan⁻¹(y) + tan⁻¹(z) = π/2, then
the value of cot⁻¹(x) + cot⁻¹(y) + cot⁻¹(z) = ?Solution
We are given: tan⁻¹x + tan⁻¹y + tan⁻¹z = π/2 We are asked to find: cot⁻¹x + cot⁻¹y + cot⁻¹z = ? We know the identity: tan⁻¹θ + cot⁻¹θ = π/2 From this identity, we can write: tan⁻¹x = π/2 - cot⁻¹x tan⁻¹y = π/2 - cot⁻¹y tan⁻¹z = π/2 - cot⁻¹z Substitute these into the given equation: (π/2 - cot⁻¹x) + (π/2 - cot⁻¹y) + (π/2 - cot⁻¹z) = π/2 Combine the π/2 terms: 3π/2 - (cot⁻¹x + cot⁻¹y + cot⁻¹z) = π/2 Rearrange the equation to solve for cot⁻¹x + cot⁻¹y + cot⁻¹z: cot⁻¹x + cot⁻¹y + cot⁻¹z = 3π/2 - π/2 cot⁻¹x + cot⁻¹y + cot⁻¹z = 2π/2 cot⁻¹x + cot⁻¹y + cot⁻¹z = π
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