Question

# The value of $\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0}$ is${\text{A}}{\text{. }}0 \\ {\text{B}}{\text{. Not Defined}} \\ {\text{C}}{\text{. }}1 \\ {\text{D}}{\text{. }} - 1 \\$

Hint: In this question we need to find the value of the given trigonometric expression. In order to evaluate it easily we will use the property that $\tan x = \cot \left( {90 - x} \right)$. This will simplify the expression and help us reach the answer.

We have been given the expression $\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0}$
Now, as we know that $\tan x = \cot \left( {90 - x} \right)$, so we will apply it to the expression starting from $\tan {50^0}$ to $\tan {85^0}$ we get,
$\tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \tan {40^0}\tan {45^0}\cot {40^0}...\cot {5^0}$
As we know that, $\tan x.\cot x = 1$

So, we get
$\Rightarrow \tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0} = 1 \cdot 1 \cdot 1 \cdot \cdot \cdot \cdot \cdot \tan {45^0}$
And as we know that$\ tan{45^0}=1$,
Hence, $\Rightarrow \tan {5^0}\tan {10^0}\tan {15^0} \cdot \cdot \cdot \cdot \tan {85^0} = 1$

Note: Whenever we face such types of problems the main point to remember is that we need to have a good grasp over trigonometric properties, some of which have been used above. We must also remember the value of tangents of some common values to use them whenever required. This helps in getting us the required condition and gets us on the right track to reach the answer.