Aptitude Question ID : 94439

The value of \tan 1^{\circ}\tan 2^{\circ}\tan 3^{\circ}........\tan 89^{\circ} is :
[A]0
[B]1
[C]\sqrt{3}
[D]\frac{1}{\sqrt{3}}

\mathbf{1}
\tan 1^{\circ}\tan 2^{\circ}\tan 3^{\circ}........\tan 89^{\circ}
= \left ( \tan 1^{\circ}\cdot \tan 89^{\circ} \right )\cdot \left ( \tan 2^{\circ}\cdot \tan 88^{\circ} \right )........\tan 45^{\circ}
= \left ( \tan 1^{\circ}\cdot \cot 1^{\circ} \right )\cdot \left ( \tan 2^{\circ}\cdot \cot 2^{\circ} \right )........\tan 45^{\circ} = 1
\left [\because  \tan \left ( 90^{\circ} - \theta \right ) = \cot \theta; \tan \theta\cdot \cot \theta = 1 \right ]
Hence option [B] is correct answer.

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