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.