sumlimits_{r = 1}^{89} {{log _3}(tan {r^circ} | vedclass

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(D) Let $S = \sum\limits_{r = 1}^{89} {{\log _3}(	an {r^\circ})}$.Using the property $\log_a x + \log_a y = \log_a (xy)$,we get $S = \log_3 (	an 1^\

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(D) Let $S = \sum\limits_{r = 1}^{89} {{\log _3}(	an {r^\circ})}$.Using the property $\log_a x + \log_a y = \log_a (xy)$,we get $S = \log_3 (	an 1^\circ \cdot 	an 2^\circ \cdot \dots \cdot 	an 89^\circ)$.We know that $	an(90^\circ - 	heta) = \cot 	heta$,so $	an 89^\circ = \cot 1^\circ$,$	an 88^\circ = \cot 2^\circ$,and so on.The product $P = 	an 1^\circ \cdot 	an 2^\circ \cdot \dots \cdot 	an 44^\circ \cdot 	an 45^\circ \cdot 	an 46^\circ \cdot \dots \cdot 	an 89^\circ$.Since $	an 	heta \cdot \cot 	heta = 1$,the product of terms $	an r^\circ \cdot 	an(90^\circ - r^\circ) = 1$ for $r = 1$ to $44$.Thus,$P = (1 \cdot 1 \cdot \dots \cdot 1) \cdot 	an 45^\circ = 1 \cdot 1 = 1$.Therefore,$S = \log_3 (1) = 0$.

$\sum\limits_{r = 1}^{89} {{\log _3}(\tan {r^\circ})} = $

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