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Question: The value of \(\tan {1^\circ}.\tan {2^\circ}.\tan {3^\circ}....................\tan {89^\circ}\) is:...

The value of tan⁡1∘.tan⁡2∘.tan⁡3∘....................tan⁡89∘\tan {1^\circ}.\tan {2^\circ}.\tan {3^\circ}....................\tan {89^\circ} is:
A) 00
B) 11
C) 22
D) 12\dfrac{1}{2}

Explanation

Solution

According to given in the question we have to determine the value of tan⁡1∘.tan⁡2∘.tan⁡3∘....................tan⁡89∘\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: ⇒tan⁡(90∘−θ)=cot⁡θ...................(A) \Rightarrow \tan ({90^\circ} - \theta ) = \cot \theta ...................(A)
⇒cot⁡θ=1tan⁡θ.................(B)\Rightarrow \cot \theta = \dfrac{1}{{\tan \theta }}.................(B)
Now, we have to convert the trigonometric terms of the given trigonometric expression,
⇒tan⁡1∘=tan⁡(90∘−89∘) ⇒tan⁡1∘=cot⁡89∘ ⇒tan⁡1∘=1tan⁡89∘ \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,
⇒tan⁡(90∘−89∘).tan⁡(90∘−88∘).tan⁡(90∘−87∘)............tan⁡87∘.tan⁡88∘.tan⁡89∘ ⇒cot⁡89∘.cot⁡88∘.cot⁡87∘.....................tan⁡87∘.tan⁡88∘.tan⁡89∘...................(1) \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.
⇒1tan⁡89∘.1tan⁡88∘.1tan⁡87∘.....................tan⁡87∘.tan⁡88∘.tan⁡89∘\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∘.tan⁡2∘.tan⁡3∘....................tan⁡89∘\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⁡θ\tan \theta to cot⁡θ\cot \theta which can be converted with the help of the formula tan⁡(90∘−θ)=cot⁡θ\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⁡θ\tan \theta to cot⁡θ\cot \theta