Question

If tan⁻¹x = $\frac{x}{10}$ for some x ∊ R, then the value of cot⁻¹ x is
$\left(\mathrm{a}\right)\frac{\mathrm{\pi }}{5}\left(\mathrm{b}\right)\frac{2\mathrm{\pi }}{5}\left(\mathrm{c}\right)\frac{3\mathrm{\pi }}{5}\left(\mathrm{d}\right)\frac{4\mathrm{\pi }}{5}$

Answer 1

Disclaimer: The solution has been provided for the following question.

If tan⁻¹x = $\frac{\mathrm{\pi }}{10}$ for some x ∊ R, then the value of cot⁻¹ x is
$\left(\mathrm{a}\right)\frac{\mathrm{\pi }}{5}\left(\mathrm{b}\right)\frac{2\mathrm{\pi }}{5}\left(\mathrm{c}\right)\frac{3\mathrm{\pi }}{5}\left(\mathrm{d}\right)\frac{4\mathrm{\pi }}{5}$

Solution:

We know

$$\begin{gathered} {\tan }^{-1}x+{\cot }^{-1}x=\frac{\mathrm{\pi }}{2},\forall x\in R \\ \Rightarrow \frac{\mathrm{\pi }}{10}+{\cot }^{-1}x=\frac{\mathrm{\pi }}{2}\left({\tan }^{-1}x=\frac{\mathrm{\pi }}{10}\right) \\ \Rightarrow {\cot }^{-1}x=\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{10} \\ \Rightarrow {\cot }^{-1}x=\frac{4\mathrm{\pi }}{10}=\frac{2\mathrm{\pi }}{5} \end{gathered}$$

Hence, the correct answer is option (b).