Question 4

cot⁻¹(cot(−11)): principal range (0, π). −11 + 4π ≈ −11 + 12.566 = 1.566, which lies in (0, π). So cot⁻¹(cot(−11)) = 4π − 11. 10 sin(2 cos⁻¹(1/√2)) = 10 sin(2·π/4) = 10 sin(π/2) = 10. 10 sin(2 tan⁻¹ 2) = 10 · (2·2)/(1+4) = 10 · 4/5 = 8. Total: 4π − 11 + 10 + 8 = 4π + 7.

**Calculating cot inverse range** I need to solve the range for cot inverse, which is straightforward between (0, π). I'm working with -11 + 4π, which equals about 1.56637 and is positive but less than π. Then there's the cosine inverse of √(1/2) at π/4 and the sine of π/2 being 1. It seems I should just give a JSON response at the end, so maybe detailed math isn't necessary. I'll stick to JSON format.

**Evaluating trigonometric terms** I'm working through some trigonometric calculations. The second term calculates to π/2, giving a sine value of 1. The next term involves sin(2 arctan(2), which simplifies down, giving a product of 8 when multiplied by 10. Then, I add it all up: 4π - 11 + 10 + 8 results in 4π + 7. I'm also cautious about arccotangent values, ensuring I stay within the principal branch. In the end, everything seems to point to option C!

For cot⁻¹(cot(-11)), since the range is (0,π), I need to shift -11 by adding 4π to get it into range, giving -11+4π ≈ 1.566, which works. Now I'm moving to the cos⁻¹ term. For the second term, cos⁻¹(1/√2) gives π/4, doubling gives π/2, so sin(π/2)=1, making that term equal to 10. For the third term, tan⁻¹(2)=θ where tanθ=2, and using the double angle formula sin(2θ)=2tanθ/(1+tan²θ)=4/5, giving 10×(4/5)=8. Adding everything up: (-11+4π)+10+8 = 4π+7, so the answer is C. Looking at this problem, I need to evaluate three terms. **Term 1:** cot⁻¹(cot(-11)) Since cot⁻¹ has range (0, π), I need to find the value in (0, π) that differs from -11 by a multiple of π. -11 + 4π ≈ -11 + 12.566 = 1.566, which lies in (0, π). So cot⁻¹(cot(-11)) = -11 + 4π **Term 2:** 10 sin(2cos⁻¹(1/√2)) cos⁻¹(1/√2) = π/4, so 2·(π/4) = π/2 sin(π/2) = 1, giving 10·1 = 10 **Term 3:** 10 sin(2tan⁻¹(2)) Let θ = tan⁻¹(2), so tan θ = 2. sin(2θ) = 2tanθ/(1+tan²θ) = 4/5 So 10·(4/5) = 8 **Sum:** (-11 + 4π) + 10 + 8 = 4π + 7 ```json