Final answer:
Simplifying cot(2 tan⁻¹(5/17)): cot(2 tan⁻¹(5/17)) = 17/5.
Step-by-step explanation:
To simplify cot(2 tan⁻¹(5/17)), we can first find the value of tan⁻¹(5/17) using the inverse tangent function. tan⁻¹(5/17) is the angle whose tangent is 5/17. Let's assume θ = tan⁻¹(5/17).
By using a right triangle, we can find the values of the opposite and adjacent sides in terms of θ. The opposite side is 5 and the adjacent side is 17.
Next, we can find the value of cot(2θ) using the cotangent function. cot(2θ) = 1/tan(2θ) = 1/(2tan(θ)/(1-tan²(θ))).
Substituting the values of the opposite and adjacent sides, we can calculate cot(2θ) = 1/(2*(5/17)/(1-(5/17)²)). Simplifying further, cot(2θ) = 17/5 or option c).