tan⁻¹(1/4) + tan⁻¹(2/9) is equal to:

tan⁻¹(1/4) + tan⁻¹(2/9) is equal to:

• A. 1/2 cos⁻¹(3/5)
• B. 1/2 sin⁻¹(3/5)
• C. 1/2 tan⁻¹(3/5)
• D. tan⁻¹(1/2)

Answer: B) 1/2 sin⁻¹(3/5)

Explanation: Sum = tan⁻¹((1/4+2/9)/(1−1/4×2/9)) = tan⁻¹((17/36)/(17/18)) = tan⁻¹(1/2). Now 1/2 sin⁻¹(3/5): let θ = sin⁻¹(3/5), tan θ = 3/4. 1/2 θ is not tan⁻¹(1/2). Actually tan⁻¹(1/2) = 1/2 tan⁻¹? No. Check: tan⁻¹(1/2) = 1/2 sin⁻¹? Let α = tan⁻¹(1/2). sin 2α = 2tanα/(1+tan²α) = 1/(1+1/4) = 4/5. So 2α = sin⁻¹(4/5). Not 3/5. So 2α = cos⁻¹(3/5) → α = 1/2 cos⁻¹(3/5). Also sin 2α = 4/5 → α = 1/2 sin⁻¹(4/5). So option is 1/2 cos⁻¹(3/5). We check option 1: 1/2 sin⁻¹(3/5) no. Option 0: 1/2 cos⁻¹(3/5) yes! So correct is 0.