Question

Find the value of ${\text{tan}}{{\text{1}}^{\text{0}}}{\text{tan}}{{\text{2}}^{\text{0}}}{\text{tan}}{{\text{3}}^{\text{0}}}........{\text{tan8}}{{\text{9}}^{\text{0}}}$as

Answer 1

Hint- To solve this question, we will use some basic formula of trigonometry. Some of them are mentioned below. And by using this we will get the value of given expression-
tan1°tan2°tan3°...tan44°tan45°tan46°...tan87°tan88°tan89°

Complete step-by-step answer:
tan1°tan2°tan3°...tan44°tan45°tan46°...tan87°tan88°tan89°
as we know,
tan$({90^0} - {\text{A)}}$ = cotA
Hence, tanA tan$({90^0} - {\text{A)}}$ = tanA cotA = tanA$ \times $$\dfrac{{\text{1}}}{{{\text{tanA}}}}$= 1.
Therefore, we can write:
${\text{tan}}{{\text{1}}^0}{\text{tan}}{{\text{2}}^{\text{0}}}{\text{tan}}{{\text{3}}^{\text{0}}}........{\text{tan8}}{{\text{9}}^{\text{0}}}$= ${\text{(tan}}{{\text{1}}^{\text{0}}}{\text{tan8}}{{\text{9}}^{\text{0}}}{\text{)(tan}}{{\text{2}}^{\text{0}}}{\text{tan8}}{{\text{8}}^{\text{0}}}{\text{)}}........{\text{(tan4}}{{\text{4}}^{\text{0}}}{\text{tan4}}{{\text{6}}^{\text{0}}}{\text{)tan4}}{{\text{5}}^{\text{0}}}$
=${\text{(tan}}{{\text{1}}^{\text{0}}}{\text{tan}}{\left( {90 - 1} \right)^{\text{0}}}{\text{)(tan}}{{\text{2}}^{\text{0}}}{\text{tan}}{\left( {90 - 2} \right)^{\text{0}}}{\text{)}}........{\text{(tan4}}{{\text{4}}^{\text{0}}}{\text{tan}}{\left( {90 - 44} \right)^{\text{0}}}{\text{)tan4}}{{\text{5}}^{\text{0}}}$
As we have,
$ \Rightarrow $tanA tan$({90^0} - {\text{A)}}$ =1
So, the above expression can written as:
=${\text{(tan}}{{\text{1}}^{\text{0}}}{\text{tan}}{\left( {90 - 1} \right)^{\text{0}}}{\text{)(tan}}{{\text{2}}^{\text{0}}}{\text{tan}}{\left( {90 - 2} \right)^{\text{0}}}{\text{)}}........{\text{(tan4}}{{\text{4}}^{\text{0}}}{\text{tan}}{\left( {90 - 44} \right)^{\text{0}}}{\text{)tan4}}{{\text{5}}^{\text{0}}}$
= $1 \times 1 \times 1 \times ........ \times 1$
= 1
Therefore, the value of ${\text{tan}}{{\text{1}}^{\text{0}}}{\text{tan}}{{\text{2}}^{\text{0}}}{\text{tan}}{{\text{3}}^{\text{0}}}........{\text{tan8}}{{\text{9}}^{\text{0}}}$ is 1.

Note- In this question key point we paired up ${\text{tan}}{{\text{1}}^{\text{0}}}{\text{tan8}}{{\text{9}}^{\text{0}}}$ and other value also like this ,so that calculation will become easy to solve. We need to remember the basic identity or formula of the trigonometry function as shown below.
sin(90°−x) = cos x
cos(90°−x) = sin x
tan(90°−x) = cot x
cot(90°−x) = tan x
sec(90°−x) = csc x
csc(90°−x) = sec x