A) 0
B) \[\sqrt{3}\]
C) \[\frac{1}{\sqrt{3}}\]
D) 1
Correct Answer: B
Solution :
\[tan\,7{}^\circ .\text{ }tan\,23{}^\circ .\text{ }tan\,60{}^\circ .\text{ }tan\,67{}^\circ .\text{ }tan\,83{}^\circ \] = tan7\[{}^\circ \].tan83\[{}^\circ \].tan23\[{}^\circ \]. \[tan\,67{}^\circ \text{ }.tan\,60{}^\circ \] \[=tan\,7{}^\circ .tan\left( 90{}^\circ -7{}^\circ \right).tan\,23{}^\circ .\] tan(90\[{}^\circ \]-23\[{}^\circ \]).tan60\[{}^\circ \] \[=tan\,7{}^\circ .\text{ }cot\,7{}^\circ .\text{ }tan\,23{}^\circ .cot\,23{}^\circ .tan\,60{}^\circ \] \[=1.1\sqrt{3}=\sqrt{3}\] \[\left[ \begin{align} & \tan (90{}^\circ -\theta )=\cot \theta : \\ & \tan \theta .\cot \theta =1 \\ \end{align} \right]\]
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