A) 0
B) 1
C) \[\infty \]
D) None of these
Correct Answer: A
Solution :
\[\log \tan 1{}^\circ +\log \tan 2{}^\circ +...+\log \tan 89{}^\circ \] \[=\]\[\log \,\,[\tan 1{}^\circ \cdot \tan 2{}^\circ ...\tan 45{}^\circ ...\tan 88{}^\circ \cdot \tan 89{}^\circ ]\] =\[\log \,\,[\tan 1{}^\circ \cdot \tan 2{}^\circ ...\tan 45{}^\circ ...\tan \] \[(90{}^\circ -2{}^\circ )\,\,\tan \,(90{}^\circ -1)]\] \[=\log \,\,[\tan 1{}^\circ \cdot cot1{}^\circ \cdot tan2{}^\circ \cdot cot2{}^\circ ...1]\] = log 1 = 0
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