A) 1
B) 3
C) 5
D) 7
Correct Answer: C
Solution :
\[\frac{\sqrt{1+\sin \,\frac{39\pi }{8}}}{\sqrt{1+\sin \,\frac{57\pi }{8}}}\,=\frac{\sqrt{1+\sin \frac{\pi }{8}}}{\sqrt{1-\sin \frac{\pi }{8}}}\] \[=\frac{\sqrt{1+\cos \frac{3\pi }{8}}}{\sqrt{1-\cos \frac{3\pi }{8}}}\,=\frac{\sqrt{2}\cos \,\frac{3\pi }{16}}{\sqrt{2-\sin \frac{3\pi }{16}}}\]\[=\cot \,\frac{3\pi }{16}=\tan \,=\frac{5\pi }{16}\] \[\therefore \] least positive value of k is 5.
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