A) \[2n\pi \pm \frac{\pi }{3},\,n\in Z\]
B) \[n\pi \pm \frac{\pi }{3},\,n\in Z\]
C) \[2n\pi \pm \frac{\pi }{6},\,n\in Z\]
D) \[n\pi \pm \frac{\pi }{6},\,n\in Z\]
Correct Answer: B
Solution :
\[\tan \theta +\tan \left( \frac{3\pi }{4}+\theta \right)=2\] \[\Rightarrow \] \[\tan \theta +\tan \left( \frac{\pi }{2}+\left( \frac{\pi }{4}+\theta \right) \right)=2\] \[\Rightarrow \] \[\tan \theta -\cot \left( \frac{\pi }{4}+\theta \right)=2\] \[\Rightarrow \] \[\tan \theta -\frac{\cot \frac{\pi }{4}\cot \theta -1}{\cos \frac{\pi }{4}+\cot \theta }=2\] \[\Rightarrow \] \[\tan \theta -\frac{\cot \theta -1}{1+\cot \theta }=2\] \[\Rightarrow \] \[\tan \theta -\frac{1-\tan \theta }{1+\tan \theta }=2\] \[\Rightarrow \] \[\tan \theta +{{\tan }^{2}}\theta -1+\tan \theta =2+2\tan \theta \] \[\Rightarrow \] \[{{\tan }^{2}}\theta =3\] \[\Rightarrow \] \[\tan \theta =\sqrt{3}=\tan \frac{\pi }{3}\] \[\Rightarrow \] \[\theta =n\pi +\frac{\pi }{3},n\in Z\]
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