Question

The value of \[\tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ\] is

 

Options

• 0

• 1

• \[\frac{1}{2}\]

   

• not defined  

Answer 1

We know that,
\[\tan\left( 90^\circ - \theta \right) = \cot\theta\]
So,

\[\tan89^\circ = \tan\left( 90^\circ - 1^\circ \right) = \cot1^\circ\]
\[\tan88^\circ = \tan\left( 90^\circ - 2^\circ \right) = \cot2^\circ\]
\[\tan87^\circ = \tan\left( 90^\circ - 3^\circ \right) = \cot3^\circ\]
 . . . .
 . . . . 
\[\tan46^\circ = \tan\left( 90^\circ - 44^\circ \right) = \cot44^\circ\]

\[\therefore \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ\]

\[ = \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan44^\circ \tan45^\circ \tan46^\circ . . . \tan87^\circ \tan88^\circ \tan89^\circ\]

\[ = \tan1^\circ \tan2^\circ \tan3^\circ . . . \tan44^\circ \tan45^\circ \cot44^\circ. . . \cot3^\circ \cot2^\circ \cot1^\circ\]

\[ = \left( \tan1^\circ\cot1^\circ \right)\left( \tan2^\circ\cot2^\circ \right) \left( \tan3^\circ\cot3^\circ \right) . . . \left( \tan44^\circ\cot44^\circ \right)\tan45^\circ\]

\[ = 1 \left( \tan45^\circ = 1\text{ and }\tan\theta\cot\theta = 1 \right)\]

Hence, the correct answer is option 1.