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Question Number 106387 by bemath last updated on 05/Aug/20

$$\cot \theta +\cot (\frac{\pi }{4}+\theta )=2$$$$\theta =?$$

Answered by 1549442205PVT last updated on 05/Aug/20

$$\cot \theta +\cot (\frac{\pi }{4}+\theta )=2$$$$\iff \frac{\sin (2\theta +\frac{\pi }{4})}{\sin \theta \sin (\frac{\pi }{4}+\theta )}=2\iff \sin (\frac{\pi }{4}+2\theta )=2\sin \theta \sin (\frac{\pi }{4}+\theta )$$$$\iff \cos 2\theta +\sin 2\theta =2\sin \theta (\cos \theta +\sin \theta )$$$$\cos 2\theta +\sin 2\theta =\sin 2\theta +{2\sin }^2\theta$$$$\iff \cos 2\theta ={2\sin }^2\theta \iff \cos 2\theta =1-\cos 2\theta$$$$\iff \cos 2\theta =\frac{1}{2}=\cos \frac{\pi }{3}\iff 2\theta =\pm \frac{\pi }{3}+2\mathrm{k}\pi$$$$\iff \mathit{\theta }=\pm \frac{\mathit{\pi }}{6}+\mathbf{k}\mathit{\pi }(\mathbf{k}\in \mathbb{Z})$$

Answered by bobhans last updated on 05/Aug/20

$$\Rightarrow \cot (\frac{\pi }{4}+\theta )=\frac{1-\tan \theta }{1+\tan \theta }=\frac{1-\frac{1}{\cot \theta }}{1+\frac{1}{\cot \theta }}$$$$=\frac{\cot \theta -1}{\cot \theta +1}.[\mathrm{let}\cot \theta =\chi ]$$$$\Rightarrow \chi +\frac{\chi -1}{\chi +1}=2;{\chi }^2+2\chi -1=2\chi +2$$$${\chi }^2=3\Rightarrow \chi =\pm \sqrt{3};\cot \theta =\pm \sqrt{3}$$$$\mathrm{or}\tan \theta =\pm \frac{1}{\sqrt{3}},\theta =\pm \frac{\pi }{6}+\mathrm{k}.\pi ★$$

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