Question

Solve the following equations:
(i) $\tan \mathrm{\theta }+\tan 2\mathrm{\theta }+\tan 3\mathrm{\theta }=0$
(ii) $\tan \mathrm{\theta }+\tan 2\mathrm{\theta }=\tan 3\mathrm{\theta }$
(iii) $\tan 3\mathrm{\theta }+\tan \mathrm{\theta }=2\tan 2\mathrm{\theta }$

Answer 1

(i)
We have:
$\tan \theta +\tan 2\theta +\tan 3\theta =0$
Now,

$$\begin{gathered} \tan \theta +\tan 2\theta +\tan (\theta +2\theta )=0 \\ \Rightarrow \tan \theta +\tan 2\theta +\left(\frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta }\right)=0 \\ \Rightarrow (\tan \theta +\tan 2\theta )(1-\tan \theta \tan 2\theta )+\tan \theta +\tan 2\theta =0 \\ \Rightarrow (\tan \theta +\tan 2\theta )(2-\tan \theta \tan 2\theta )=0 \end{gathered}$$
$\Rightarrow \tan \theta +\tan 2\theta =0$ or$2-\tan \theta \tan 2\theta =0$

Now,
$$\begin{gathered} \tan \theta +\tan 2\mathrm{\theta }=0 \\ \Rightarrow \tan \theta =-\tan 2\theta \\ \Rightarrow \tan \theta =\tan -2\theta \\ \Rightarrow \theta =n\pi -2\theta \\ \Rightarrow 3\theta =n\pi \\ \Rightarrow \theta =\frac{n\mathrm{\pi }}{3},n\in Z \end{gathered}$$
And,

$$\begin{gathered} 2-\tan \theta \tan 2\theta =0 \\ \Rightarrow \tan \theta \tan 2\theta =2 \\ \Rightarrow \frac{\sin \theta }{\cos \theta }\frac{\sin 2\theta }{\cos 2\theta }=2 \\ \Rightarrow \frac{2{\sin }^2\theta \cos \theta }{\cos \theta }=2{\cos }^2\theta -2{\sin }^2\theta \\ \Rightarrow 4{\sin }^2\theta =2{\cos }^2\theta \\ \Rightarrow {\tan }^2\theta =\frac{1}{2}\Rightarrow {\tan }^2\theta ={\tan }^2\alpha \\ \Rightarrow \theta =m\pi +\alpha ,m\in Z,\alpha ={\tan }^{-1}\left(\frac{1}{2}\right) \end{gathered}$$

∴ $\theta =\frac{n\mathrm{\pi }}{3},n\in Z$ or $\theta =m\pi +\alpha ,m\in Z$
Here,
$\alpha ={\tan }^{-1}\left(\frac{1}{2}\right)$

(ii)
Given:
$\tan \theta +\tan 2\theta =\tan 3\theta$
Now,
$$\begin{gathered} \tan \theta +\tan 2\theta =\tan (\theta +2\theta ) \\ \Rightarrow \tan \theta +\tan 2\theta =\frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta } \\ \Rightarrow \tan \theta +\tan 2\theta -\frac{\tan \theta +\tan 2\theta }{1-\tan \theta \tan 2\theta }=0 \\ \Rightarrow (\tan \theta +\tan 2\theta )(1-\tan \theta \tan 2\theta )-(\tan \theta +\tan 2\theta )=0 \\ \Rightarrow (\tan \theta +\tan 2\theta )(1-\tan \theta \tan 2\theta -1)=0 \\ \Rightarrow (\tan \theta +\tan 2\theta )(-\tan \theta \tan 2\theta )=0 \end{gathered}$$

$\Rightarrow \tan \theta +\tan 2\theta =0$ or $\tan \theta \tan 2\theta =0$
Now,
$$\begin{gathered} \tan \theta +\tan 2\theta =0 \\ \Rightarrow \tan \theta =-\tan 2\theta \\ \Rightarrow \tan \theta =\tan -2\theta \\ \Rightarrow \theta =n\pi -2\theta ,n\in Z \\ \Rightarrow 3\theta =n\pi \\ \Rightarrow \theta =\frac{n\mathrm{\pi }}{3},n\in Z \end{gathered}$$
And,

$$\begin{gathered} \tan \theta \tan 2\theta =0 \\ \Rightarrow \frac{\sin \theta }{\cos \theta }\frac{\sin 2\theta }{\cos 2\theta }=0 \\ \Rightarrow \frac{2{\sin }^2\theta }{{\cos }^2\theta -{\sin }^2\theta }=0 \\ \Rightarrow {\sin }^2\theta =0\Rightarrow {\sin }^2\theta ={\sin }^{2}0 \\ \Rightarrow \theta =m\pi ,m\in Z \end{gathered}$$

∴ $\theta =\frac{n\mathrm{\pi }}{3},n\in Z$ or $\theta =m\pi ,m\in Z$

(iii)
Given:
$\tan 3\theta +\tan \theta =2\tan 2\theta$
Now,
$$\begin{gathered} \tan 3\theta -\tan 2\theta =\tan 2\theta -\tan \theta \\ \Rightarrow \tan \theta (1+\tan 3\theta \tan 2\theta )=\tan \theta (1+\tan 2\theta \tan \theta )\left[\tan \left(A-B\right)=\frac{\tan A-\tan B}{1+\tan A\tan B}\right] \\ \Rightarrow \tan \theta (1+\tan 3\theta \tan 2\theta -1-\tan 2\theta \tan \theta )=0 \\ \Rightarrow \tan \theta \tan 2\theta (\tan 3\theta -\tan \theta )=0 \end{gathered}$$

$\Rightarrow \tan 2\theta =0$ or, $\tan \theta =0$ or, $\tan 3\theta -\tan \theta =0$
And,
$\tan 2\theta =0\Rightarrow 2\theta =n\pi \Rightarrow \theta =\frac{n\mathrm{\pi }}{2},n\in Z$
Or,
$\tan 3\theta -\tan \theta =0\Rightarrow \tan 3\theta =\tan \theta \Rightarrow 3\theta =n\pi +\theta \Rightarrow 2\theta =n\mathrm{\pi }\Rightarrow \mathrm{\theta }=\frac{\mathrm{n\pi }}{2},\mathrm{n}\in \mathrm{Z}$
And,
$\tan \theta =0\Rightarrow \theta =m\pi ,m\in Z$

∴ $\theta =\frac{n\mathrm{\pi }}{2},n\in Z$ or $\theta =m\mathrm{\pi },\mathrm{m}\in \mathrm{Z}$