Question

Find the value of the expression \[\tan 1 \cdot \tan 2 \cdot \tan 3 \ldots \cdot \tan 89\].

Answer 1

Hint: We will try to convert tan to its reciprocal value that is cot so that it will easily cancel out with each other and we will get a simplest value that will help us to get the final answer easily. After doing this we will get a finite value left to us which will be the final answer to this question.
 $ \tan (90 - x) = \cot x $ where, $ x $ is angle in degrees.
 $ \tan x \times \cot x = 1 $

Complete step-by-step answer:
The given expression is:
\[\tan 1 \cdot \tan 2 \cdot \tan 3 \ldots \cdot \tan 89\]
\[ = \tan 1 \cdot \tan 2 \cdot \tan 3 \ldots \cdot \tan 87 \cdot \tan 88 \cdot \tan 89\]
\[ = \tan 1 \cdot \tan 2 \cdot \tan 3 \ldots \cdot \tan (90 - 3) \cdot \tan (90 - 2) \cdot \tan (90 - 1)\]
\[ = \tan 1 \cdot \tan 2 \cdot \tan 3 \ldots \cdot \cot 3 \cdot \cot 2 \cdot \cot 1\] [ $ \because \tan (90 - x) = \cot x $ ]
Rearranging the above expression we get:
\[ = \tan 1 \cdot \cot 1 \cdot \tan 2 \cdot \cot 2 \cdot \tan 3 \cdot \cot 3 \ldots \cdot \tan 45\]
\[ = (\tan 1 \cdot \cot 1) \cdot (\tan 2 \cdot \cot 2) \cdot (\tan 3 \cdot \cot 3) \ldots \cdot (\tan 45)\]
\[ = 1 \cdot (\tan 45) = \tan 45\] [ $ \because \tan x \times \cot x = 1 $ ]
\[ = 1\] [ $ \because \tan 45 = 1 $ ]
Therefor the value of \[\tan 1 \cdot \tan 2 \cdot \tan 3 \ldots \cdot \tan 89\] is \[1\]

Note: Notice carefully that we are converting tan angles to cot angles up to a certain terms so that they cancel with each other. We should remember all the trigonometry values and functions so that easily we can get these answers.