Direct Composition of $$\cos$$ and $$\cos^{-1}$$

Rarely Tested

Direct Composition of $$\cos$$ and $$\cos^{-1}$$

## Formula

$$\cos(\cos^{-1}x)=x,\quad -1\le x\le1$$

## Usage

- Used to eliminate a direct cosine function applied to its inverse.

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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