log sin 1^{circ} cdot log sin 2^{circ} cdot l | vedclass

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Question

(A) The given expression is a product of terms of the form $\log \sin 	heta^{\circ}$ for $	heta = 1, 2, 3, \ldots, 179$.The expression is: $P = \log

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(A) The given expression is a product of terms of the form $\log \sin 	heta^{\circ}$ for $	heta = 1, 2, 3, \ldots, 179$.The expression is: $P = \log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \cdot \ldots \cdot \log \sin 90^{\circ} \cdot \ldots \cdot \log \sin 179^{\circ}$.We know that $\sin 90^{\circ} = 1$.Therefore,the term corresponding to $	heta = 90^{\circ}$ is $\log \sin 90^{\circ} = \log 1$.Since $\log 1 = 0$,the entire product becomes zero because one of the factors is zero.Thus,$P = \log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \cdot \ldots \cdot (0) \cdot \ldots \cdot \log \sin 179^{\circ} = 0$.

$\log \sin 1^{\circ} \cdot \log \sin 2^{\circ} \cdot \log \sin 3^{\circ} \cdot \ldots \cdot \log \sin 179^{\circ} = ?$

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