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  4. The solution of tan 2θ: tan θ = 1 is

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Q. The solution of $\tan 2\theta\: \tan \theta = 1$ is

Trigonometric Functions

Solution:

$\tan\,2 \theta \,\tan\,\theta=1$
$\Rightarrow \,\frac{2\,\tan\,\theta}{1-\tan^2\,\theta}\tan\, \theta=1$
$\Rightarrow 2\,\tan^2\,\theta=1-\tan^2\theta$
$\Rightarrow 3\,\tan^2\,\theta=1$
$\Rightarrow \,\tan\,\theta=\pm\,\frac{1}{\sqrt{3}}=\tan\,\left(\pm\frac{\pi}{6}\right)$
$\Rightarrow \, \theta=n\pi\,\pm\frac{pi}{6}(m\,\in\,Z)$
= $(6n \,\pm\,1)\frac{\pi}{6}$