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Question: Calculate \(\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.....................\tan {89^\circ }.\)...

Calculate tan⁡1∘.tan⁡2∘.tan⁡3∘.....................tan⁡89∘.\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.....................\tan {89^\circ }.

Explanation

Solution

In this question first try to convert like tan⁡1∘.tan⁡2∘.tan⁡3∘........tan⁡45∘...........tan⁡(90∘−3∘).tan⁡(90∘−2∘).tan⁡(90∘−1∘)\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\tan ({90^\circ } - {3^\circ }).\tan ({90^\circ } - {2^\circ }).\tan ({90^\circ } - {1^\circ }) then use formula tan⁡(90∘−θ)=cot⁡θ\tan ({90^\circ } - \theta ) = \cot \theta and tan⁡θ.cot⁡θ=1\tan \theta .\cot \theta = 1 from these we will proceed to the result .

Complete step-by-step answer:
So in this we have to find the value of tan⁡1∘.tan⁡2∘.tan⁡3∘.....................tan⁡89∘.\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.....................\tan {89^\circ }.
we know that the tan⁡(90∘−θ)=cot⁡θ\tan ({90^\circ } - \theta ) = \cot \theta So from this
tan⁡1∘.tan⁡2∘.tan⁡3∘........tan⁡45∘...........tan⁡87∘.tan⁡88∘.tan⁡89∘.\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\tan {87^\circ }.\tan {88^\circ }.\tan {89^\circ }.
we can write it as
tan⁡1∘.tan⁡2∘.tan⁡3∘........tan⁡45∘...........tan⁡(90∘−3∘).tan⁡(90∘−2∘).tan⁡(90∘−1∘)\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\tan ({90^\circ } - {3^\circ }).\tan ({90^\circ } - {2^\circ }).\tan ({90^\circ } - {1^\circ })
So from here tan⁡(90∘−θ)=cot⁡θ\tan ({90^\circ } - \theta ) = \cot \theta hence ,
tan⁡1∘.tan⁡2∘.tan⁡3∘........tan⁡45∘...........cot⁡3∘.cot⁡2∘.cot⁡1∘\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\cot {3^\circ }.\cot {2^\circ }.\cot {1^\circ }
Since we know that the tan⁡θ.cot⁡θ=1\tan \theta .\cot \theta = 1
Hence the term
tan⁡1∘.cot⁡1∘=1\tan {1^\circ }.\cot {1^\circ } = 1
tan⁡2∘.cot⁡2∘=1\tan {2^\circ }.\cot {2^\circ } = 1
tan⁡3∘.cot⁡3∘=1\tan {3^\circ }.\cot {3^\circ } = 1
.............
Similarly 4444 pairs are found and have value is equal to 11 one term is remaining that is tan⁡45∘\tan {45^\circ }
we know that tan⁡45∘=1\tan {45^\circ } = 1
tan⁡1∘.tan⁡2∘.tan⁡3∘.cot⁡3∘.cot⁡2∘.cot⁡1∘.........tan⁡45∘\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.\cot {3^\circ }.\cot {2^\circ }.\cot {1^\circ }.........\tan {45^\circ }
On putting the values we get
1.1.1.........1...1.1.1.........1...
Hence it is equal to 11
The value of tan⁡1∘.tan⁡2∘.tan⁡3∘.....................tan⁡89∘=1\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.....................\tan {89^\circ } = 1

Note: This question will be also framed like cot⁡1∘.cot⁡2∘.cot⁡3∘.....................cot⁡89∘\cot {1^\circ }.\cot {2^\circ }.\cot {3^\circ }.....................\cot {89^\circ } solve it as we solve above. The answer is also the same.In this two properties will use that is tan⁡θ.cot⁡θ=1\tan \theta .\cot \theta = 1 and cot⁡(90∘−θ)=tan⁡θ\cot ({90^\circ } - \theta ) = \tan \theta .Students should remember the important trigonometric identities and formulas for solving these types of questions.