A) \[2n\pi +\frac{\pi }{4}\]
B) \[2n\pi \pm \frac{\pi }{4}\]
C) \[2n\pi -\frac{\pi }{4}\]
D) \[n\pi -\frac{\pi }{4}\]
E) \[n\pi +\frac{\pi }{4}\]
Correct Answer: E
Solution :
\[\sin \left( \frac{\pi }{4}\cot \theta \right)=\cos \left( \frac{\pi }{4}\tan \theta \right)\] \[\Rightarrow \] \[\sin \left( \frac{\pi }{4}\cot \theta \right)=\sin \left( \frac{\pi }{2}-\frac{\pi }{4}\tan \theta \right)\] \[\Rightarrow \] \[\frac{\pi }{4}(\tan \theta +\cot \theta )=\frac{\pi }{2}\] \[\Rightarrow \] \[{{\tan }^{2}}\theta +1-2\tan \theta =0\] \[\Rightarrow \] \[{{(\tan \theta -1)}^{2}}=0\] \[\Rightarrow \] \[\tan \theta =1=\tan \frac{\pi }{4}\] \[\Rightarrow \] \[\theta =n\pi +\frac{\pi }{4}\]
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