Question

The value of $\tan {1^\circ}.\tan {2^\circ}.\tan {3^\circ}....................\tan {89^\circ}$ is:
A) $0$
B) $1$
C) $2$
D) $\dfrac{1}{2}$

Answer 1

Hint: According to given in the question we have to determine the value of $\tan {1^\circ}.\tan {2^\circ}.\tan {3^\circ}....................\tan {89^\circ}$ so, to solve the given trigonometric expression we have to use the formula as mentioned below:

Formula used: $ \Rightarrow \tan ({90^\circ} - \theta ) = \cot \theta ...................(A)$
$ \Rightarrow \cot \theta = \dfrac{1}{{\tan \theta }}.................(B)$
Now, we have to convert the trigonometric terms of the given trigonometric expression,
$
   \Rightarrow \tan {1^\circ} = \tan ({90^\circ} - {89^\circ}) \\
   \Rightarrow \tan {1^\circ} = \cot {89^\circ} \\
   \Rightarrow \tan {1^\circ} = \dfrac{1}{{\tan {{89}^\circ}}}
 $
Now, same as we have to convert all the remaining trigonometric terms and eliminate the terms which can be eliminated.

Complete step-by-step solution:
Step 1: First of all we have to use the formula (A) as mentioned in the solution hint to convert the terms in the given trigonometric expression,
$
   \Rightarrow \tan ({90^\circ} - {89^\circ}).\tan ({90^\circ} - {88^\circ}).\tan ({90^\circ} - {87^\circ})............\tan {87^\circ}.\tan {88^\circ}.\tan {89^\circ} \\
   \Rightarrow \cot {89^\circ}.\cot {88^\circ}.\cot {87^\circ}.....................\tan {87^\circ}.\tan {88^\circ}.\tan {89^\circ}...................(1)
 $
Step 2: Now, to solve the expression (1) above we have to use the formula (B) as mentioned in the solution hint.
$ \Rightarrow \dfrac{1}{{\tan {{89}^\circ}}}.\dfrac{1}{{\tan {{88}^\circ}}}.\dfrac{1}{{\tan {{87}^\circ}}}.....................\tan {87^\circ}.\tan {88^\circ}.\tan {89^\circ}$...…………….(2)
Step 3: Now, on eliminating all the terms as obtained in the expression (2),
= 1
Final solution: Hence, with the help of formula (A) and (B) as mentioned in the solution hint we have obtained the value of $\tan {1^\circ}.\tan {2^\circ}.\tan {3^\circ}....................\tan {89^\circ}$= 1.

Therefore option (B) is correct.

Note: It is necessary to convert the trigonometric terms such as $\tan \theta $to $\cot \theta $which can be converted with the help of the formula $\tan ({90^\circ} - \theta ) = \cot \theta $
It is necessary to eliminate the terms in the obtained trigonometric expression which can be eliminate after converting $\tan \theta $to $\cot \theta $