A) \[\tan 3\theta \]
B) \[\cot 3\theta \]
C) \[\tan 6\theta \]
D) \[\cot 6\theta \]
Correct Answer: C
Solution :
\[\frac{\sin \,\,3\theta +\sin \,\,5\theta +\sin \,7\theta +\sin 9\theta }{\cos 3\theta +\cos 5\theta +\cos \,7\theta +\cos \,9\theta }\] \[=\frac{(\sin \,3\theta +\sin \,9\theta )+(\sin \,5\theta +\sin \,7\theta )}{(\cos \,3\theta +\cos \,9\theta )+(\cos \,5\theta +\cos \,7\theta )}\] \[=\frac{2\,\sin \,6\theta \,\cos \,3\theta +2\,\sin \,6\theta \,\cos \,\theta }{2\,\cos \,6\theta \,\cos \,3\theta +2\,\cos \,6\theta \,\cos \,\theta }\] \[=\frac{2\,\sin \,6\theta \,(\cos \,3\theta +\cos \theta )}{2\,\cos \,6\theta \,(\cos \,3\theta +\cos \theta )}=\tan \,6\theta \].
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