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