Question

${\cos }^{-1}\left(x\right)={\cot }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ where x is in the common domain of the functions.

• A. True
• B. False

Answer 1

The correct option is A True

Let ${\cos }^{-1}x=\theta$ .......$\left(1\right)$

$\Rightarrow \cos \theta =x$

where $x$ is in the common domain of the functions.

$\Rightarrow adjacentside=x$ and $hypotenuse=1$

$\Rightarrow oppositeside=\sqrt{{\left(hypotenuse\right)}^2-{\left(adjacentside\right)}^2}=\sqrt{1-x^2}$

$\therefore \cot \theta =\frac{x}{\sqrt{1-x^2}}$

$\therefore \theta ={\cot }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$

$\therefore {\cos }^{-1}x={\cot }^{-1}\left(\frac{x}{\sqrt{1-x^2}}\right)$ from $\left(1\right)$