Principal-Value Rule for $$\cot^{-1}(\cot x)$$

Rarely Tested

Principal-Value Rule for $$\cot^{-1}(\cot x)$$

## Formula

The value of:

$$\cot^{-1}(\cot x)$$

is the unique angle in:

$$(0,\pi)$$

coterminal with $$x$$ modulo $$\pi$$.

## Usage

- Used to correctly simplify inverse cotangent of cotangent for arbitrary real $$x$$.

Question 1

Considering only the principal values of the inverse trigonometric functions, the value of

$$\cot^{-1}(\cot(-11))+10\,\sin\!\left(2\cos^{-1}\!\left(\tfrac{1}{\sqrt{2}}\right)\right)+10\sin(2\tan^{-1}(2))$$

is

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