Final answer:
To solve the trigonometric expression, one must determine the angles from the inverse cotangent and secant functions, and then apply the sine difference formula to find the sine of the difference between those angles.
Step-by-step explanation:
The task is to solve the trigonometric expression: sin(cot⁻¹(2/3) - sec⁻¹(3)). We need to find the angles corresponding to the given inverse trigonometric functions, and then calculate the sine of the difference between those angles. Let x = cot⁻¹(2/3). Then cot(x) = 2/3, which corresponds to a right-angled triangle with adjacent side 2, opposite side 3, and the hypotenuse √(22 + 32) = √13.
Similarly, let y = sec⁻¹(3). Then sec(y) = 3, which corresponds to a right-angled triangle with adjacent side 1, the hypotenuse 3, and opposite side √(32 - 12) = √8. Now, using the sine of a difference formula sin(a - b) = sin(a)cos(b) - cos(a)sin(b), we can find sin(x - y).
The final numerical calculation will reveal which of the provided options matches the result.