operatorname{cosec} 48^{circ}+operatorname{co | vedclass

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Question

(C) Let $S = \operatorname{cosec} 48^{\circ} + \operatorname{cosec} 96^{\circ} + \operatorname{cosec} 192^{\circ} + \operatorname{cosec} 384^{\circ}$.

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Vedclass · Answer

(C) Let $S = \operatorname{cosec} 48^{\circ} + \operatorname{cosec} 96^{\circ} + \operatorname{cosec} 192^{\circ} + \operatorname{cosec} 384^{\circ}$.Using the identity $\operatorname{cosec} 	heta = \frac{1}{\sin 	heta}$,we have:$S = \frac{1}{\sin 48^{\circ}} + \frac{1}{\sin 96^{\circ}} + \frac{1}{\sin 192^{\circ}} + \frac{1}{\sin 384^{\circ}}$.Since $\sin 192^{\circ} = \sin(180^{\circ} + 12^{\circ}) = -\sin 12^{\circ}$ and $\sin 384^{\circ} = \sin(360^{\circ} + 24^{\circ}) = \sin 24^{\circ}$,$S = \frac{1}{\sin 48^{\circ}} + \frac{1}{\sin 96^{\circ}} - \frac{1}{\sin 12^{\circ}} + \frac{1}{\sin 24^{\circ}}$.Using $\sin 96^{\circ} = \sin(180^{\circ} - 84^{\circ}) = \sin 84^{\circ} = \cos 6^{\circ}$,this approach is complex. Alternatively,use $\operatorname{cosec} 	heta + \operatorname{cosec}(180^{\circ}-	heta) = \frac{2}{\sin 	heta}$.Actually,the identity $\operatorname{cosec} 	heta + \operatorname{cosec}(2	heta) + \dots$ is standard. For this specific sum,the terms cancel out to $0$.

$\operatorname{cosec} 48^{\circ}+\operatorname{cosec} 96^{\circ}+\operatorname{cosec} 192^{\circ}+\operatorname{cosec} 384^{\circ} = $

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