Question

Write the value of ${\cot }^{-1}(-x)$ for all $xϵR$ in terms of ${\cot }^{-1}x$.

Answer 1

Let ${\cot }^{-1}(-x)=\theta$

$\Rightarrow \cot \theta =-x$

$\Rightarrow \cot (\pi -\theta )=x$

$\Rightarrow \pi -\theta ={\cot }^{-1}x$

$\Rightarrow \theta =\pi -{\cot }^{-1}x$

$\therefore {\cot }^{-1}(-x)=\pi -{\cot }^{-1}x$ for $x\in R$

Hence proved.