Evaluate $\tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $

Hint: Here, we are required to find the value of $\tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $. Hence, we will use the trigonometric identities of the relationship between $\tan \theta $ and $\cot \theta $. Using those identities in this question, we will be able to simplify it further. Also, using the trigonometric tables, we can substitute the value of $\tan 45^\circ $. Thus, this will help us to evaluate the given trigonometric expression where various tangent angles are multiplying with each other.

Formula Used:
1.$\tan \theta = \cot \left( {90^\circ - \theta } \right)$
2.$\cot \theta = \dfrac{1}{{\tan \theta }}$

Complete step-by-step answer:
To find: $\tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $
Now, we will the formula: $\tan \theta = \cot \left( {90^\circ - \theta } \right)$
Here, substituting $\theta = 10^\circ $
$ \Rightarrow \tan 10^\circ = \cot \left( {90^\circ - 10^\circ } \right) = \cot 80^\circ $
Again, substituting $\theta = 20^\circ $
$ \Rightarrow \tan 20^\circ = \cot \left( {90^\circ - 20^\circ } \right) = \cot 70^\circ $
And, substituting $\theta = 40^\circ $
$ \Rightarrow \tan 40^\circ = \cot \left( {90^\circ - 40^\circ } \right) = \cot 50^\circ $
Hence, substituting the values of $\tan 10^\circ ,\tan 20^\circ $ and $\tan 40^\circ $in the question, we get,
$\tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $$ = \cot 80^\circ \cdot \cot 70^\circ \cdot \cot 50^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $
Now, we know that,
$\cot \theta = \dfrac{1}{{\tan \theta }}$
Hence, using this formula, we get,
$ \Rightarrow \tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $$ = \dfrac{1}{{\tan 80^\circ }} \cdot \dfrac{1}{{\tan 70^\circ }} \cdot \dfrac{1}{{\tan 50^\circ }} \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $
$ \Rightarrow \tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $$ = \left( {\dfrac{1}{{\tan 80^\circ }} \times \tan 80^\circ } \right) \cdot \left( {\dfrac{1}{{\tan 70^\circ }} \times \tan 70^\circ } \right) \cdot \left( {\dfrac{1}{{\tan 50^\circ }} \times \tan 50^\circ } \right) \cdot \tan 45^\circ $
Also, using the trigonometric tables, we know that, $\tan 45^\circ = 1$
$ \Rightarrow \tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ = 1 \times 1 \times 1 \times 1$
$ \Rightarrow \tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ = 1$
Thus, the value of $\tan 10^\circ \cdot \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 45^\circ \cdot \tan 50^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ $is 1.
This is the required answer.

Note: This question involved Trigonometry which is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In the simple terms they are written as ‘sin’, ‘cos’ and ‘tan’.