Question

Show that${\tan }^{-1}v+co{t}^{-1}v=\frac{\mathrm{\pi }}{2}$

Short answer

The equation ${\tan }^{-1}v+co{t}^{-1}v=\frac{\mathrm{\pi }}{2}$can be obtained as

$$\begin{gathered} x={\tan }^{-1}\left(v\right) \\ y=\mathrm{co}{t}^{-1}\left(v\right) \\ so\tan \left(x\right)=v \\ \mathrm{co}t\left(y\right)=v \\ \tan \left(x\right)=\mathrm{co}t\left(y\right) \\ \tan \left(x\right)=\sin \left(\frac{\mathrm{\pi }}{2}-y\right) \\ x=\frac{\mathrm{\pi }}{2}-y \\ x+y=\frac{\mathrm{\pi }}{2} \\ {\tan }^{-1}\left(v\right)+\mathrm{co}{t}^{-1}\left(v\right)=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Step-by-step solution

Step 1. Given data 

The given equation for verification is

${\tan }^{-1}v+co{t}^{-1}v=\frac{\mathrm{\pi }}{2}$

Step 2. Considering variables

Consider

$$\begin{gathered} x={\tan }^{-1}\left(v\right) \\ y=co{t}^{-1}\left(v\right) \end{gathered}$$

Change inverse trigonometric into normal trigonometric function

$$\begin{gathered} \tan \left(x\right)=v \\ \mathrm{co}t\left(y\right)=v \end{gathered}$$

so$\tan \left(x\right)=\mathrm{co}t\left(y\right)$

Step 3. proof 

Use formula

$\mathrm{co}t\left(n\right)=\tan (\frac{\mathrm{\pi }}{2}-n)$

so$$\begin{gathered} \tan \left(x\right)=\mathrm{co}t\left(y\right) \\ \tan \left(x\right)=\tan \left(\frac{\mathrm{\pi }}{2}-y\right) \\ x=\frac{\mathrm{\pi }}{2}-y \\ x+y=\frac{\mathrm{\pi }}{2} \end{gathered}$$

Substitute expression of xand y

$$\begin{gathered} x+y=\frac{\mathrm{\pi }}{2} \\ {\tan }^{-1}v+co{t}^{-1}v=\frac{\mathrm{\pi }}{2} \end{gathered}$$