Trigonometric functions are mathematical functions that relate to angles in a right-angled triangle. The most common trigonometric functions are sine, cosine, and tangent. They are abbreviated as sin, cos, and tan.


The tan inverse function is the inverse of the tangent function. The notation for the tan inverse function is tan^-1, or arctan. The tan inverse function takes an input value and returns an angle in radians whose tangent is equal to the input value.


The cot inverse function is the inverse of the cotangent function. The notation for the cot inverse function is cot^-1, or arccot. The cot inverse function takes an input value and returns an angle in radians whose cotangent is equal to the input value.


tan inverse root3 = pi/3 (In radians)
cot inverse -root 3 = 5pi/6 (In radians)

Therefore,
tan inverse root3 - cot inverse -root 3 = pi/3 - 5pi/6

To solve this expression, we need to find a common denominator for pi/3 and 5pi/6. The common denominator is 6.

Therefore,
pi/3 - 5pi/6 = 2pi/6 - 5pi/6 = -3pi/6 = -pi/2


tan inverse root3 - cot inverse -root 3 = -pi/2

Therefore, the final answer is -pi/2.