Question

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

Answer 1

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

Complete step-by-step answer:
So in this we have to find the value of $\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.....................\tan {89^\circ }.$
we know that the $\tan ({90^\circ } - \theta ) = \cot \theta $ So from this
$\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\tan {87^\circ }.\tan {88^\circ }.\tan {89^\circ }.$
we can write it as
$\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\tan ({90^\circ } - {3^\circ }).\tan ({90^\circ } - {2^\circ }).\tan ({90^\circ } - {1^\circ })$
So from here $\tan ({90^\circ } - \theta ) = \cot \theta $ hence ,
$\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }........\tan {45^\circ }...........\cot {3^\circ }.\cot {2^\circ }.\cot {1^\circ }$
Since we know that the $\tan \theta .\cot \theta = 1$
Hence the term
$\tan {1^\circ }.\cot {1^\circ } = 1$
$\tan {2^\circ }.\cot {2^\circ } = 1$
$\tan {3^\circ }.\cot {3^\circ } = 1$
.............
Similarly $44$ pairs are found and have value is equal to $1$ one term is remaining that is $\tan {45^\circ }$
we know that $\tan {45^\circ } = 1$
$\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.\cot {3^\circ }.\cot {2^\circ }.\cot {1^\circ }.........\tan {45^\circ }$
On putting the values we get
$1.1.1.........1...$
Hence it is equal to $1$
The value of $\tan {1^\circ }.\tan {2^\circ }.\tan {3^\circ }.....................\tan {89^\circ } = 1$


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