Question

$\tan \theta =\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+}}}}}}$
What is $\theta =?$

Answer 1

$\tan \theta =\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+...}}}}}$

$⟹\tan \theta =\frac{1}{2+\tan \theta }$

$⟹2\tan \theta +{\tan }^2\theta =1$

$⟹{\tan }^2\theta +2\tan \theta -1=0$

$⟹\tan \theta =\frac{-2\pm \sqrt{2^2+4}}{2}$

$⟹\tan \theta =\frac{-2\pm \sqrt{8}}{2}$

$⟹\tan \theta =\frac{-2\pm 2\sqrt{2}}{2}$

$⟹\tan \theta =-1\pm \sqrt{2}$

$⟹\tan \theta =\sqrt{2}-1=\tan 22{\frac{1}{2}}^{\circ }$

$\therefore \theta =22{\frac{1}{2}}^{\circ }$