Question

Evaluate each of the following:

(i) ${\cot }^{-1}\frac{1}{\sqrt{3}}-{\mathrm{cosec}}^{-1}\left(-2\right)+{\sec }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

(ii) ${\cot }^{-1}\left\{2\cos \left({\sin }^{-1}\frac{\sqrt{3}}{2}\right)\right\}$

(iii) ${\mathrm{cosec}}^{-1}\left(-\frac{2}{\sqrt{3}}\right)+2{\cot }^{-1}\left(-1\right)$

(iv) ${\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{1}{\sqrt{3}}\right)+{\tan }^{-1}\left(\sin \left(-\frac{\mathrm{\pi }}{2}\right)\right)$

Answer 1

(i)
$$\begin{gathered} {\cot }^{-1}\frac{1}{\sqrt{3}}-{\mathrm{cosec}}^{-1}\left(-2\right)+{\sec }^{-1}\left(\frac{2}{\sqrt{3}}\right)={\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right)-{\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{6}\right)\right]+{\sec }^{-1}\left(\sec \frac{\mathrm{\pi }}{6}\right) \\ =\frac{\mathrm{\pi }}{3}+\frac{\mathrm{\pi }}{6}+\frac{\mathrm{\pi }}{6} \\ =\frac{2\mathrm{\pi }}{3} \end{gathered}$$

(ii)
$$\begin{gathered} {\cot }^{-1}\left\{2\cos \left({\sin }^{-1}\frac{\sqrt{3}}{2}\right)\right\}={\cot }^{-1}\left\{2\cos \left[{\sin }^{-1}\left(\sin \frac{\mathrm{\pi }}{3}\right)\right]\right\} \\ ={\cot }^{-1}\left(2\cos \frac{\mathrm{\pi }}{3}\right) \\ ={\cot }^{-1}\left(2\times \frac{1}{2}\right) \\ ={\cot }^{-1}\left(1\right) \\ ={\cot }^{-1}\left(\tan \frac{\mathrm{\pi }}{4}\right) \\ =\frac{\mathrm{\pi }}{4} \end{gathered}$$

(iii)
$$\begin{gathered} {\mathrm{cosec}}^{-1}\left(-\frac{2}{\sqrt{3}}\right)+2{\cot }^{-1}\left(-1\right)={\mathrm{cosec}}^{-1}\left[\mathrm{cosec}\left(-\frac{\mathrm{\pi }}{3}\right)\right]+2{\cot }^{-1}\left[\cot \left(\frac{3\mathrm{\pi }}{4}\right)\right] \\ =-\frac{\mathrm{\pi }}{3}+2\times \frac{3\mathrm{\pi }}{4} \\ =-\frac{\mathrm{\pi }}{3}+\frac{3\mathrm{\pi }}{2} \\ =\frac{7\mathrm{\pi }}{6} \end{gathered}$$

(iv)
$$\begin{gathered} {\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)+{\cot }^{-1}\left(\frac{1}{\sqrt{3}}\right)+{\tan }^{-1}\left(\sin \left(-\frac{\mathrm{\pi }}{2}\right)\right)={\tan }^{-1}\left[\tan \left(-\frac{\mathrm{\pi }}{6}\right)\right]+{\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right)+{\tan }^{-1}\left(-1\right) \\ ={\tan }^{-1}\left[\tan \left(-\frac{\mathrm{\pi }}{6}\right)\right]+{\cot }^{-1}\left(\cot \frac{\mathrm{\pi }}{3}\right)+{\tan }^{-1}\left[\tan \left(-\frac{\mathrm{\pi }}{4}\right)\right] \\ =-\frac{\mathrm{\pi }}{6}+\frac{\mathrm{\pi }}{3}-\frac{\mathrm{\pi }}{4} \\ =-\frac{\mathrm{\pi }}{12} \end{gathered}$$