The question asks for the value of the expression: $(\tan 2^\circ \tan 88^\circ) (\tan 3^\circ \tan 87^\circ) ... (\tan 43^\circ \tan 47^\circ) \tan 45^\circ$

Key Trigonometric Identity

We utilize the co-function identity for tangent: $ \tan(90^\circ - \theta) = \cot(\theta) $ And the reciprocal identity: $ \cot(\theta) = \frac{1}{\tan(\theta)} $ Combining these gives: $ \tan(90^\circ - \theta) = \frac{1}{\tan(\theta)} $ Therefore, $ \tan(\theta) \tan(90^\circ - \theta) = 1 $

Applying the Identity to Pairs

The expression contains pairs of tangent functions whose angles sum to $90^\circ$. Let's examine these pairs:

• For the first pair: $\tan 2^\circ \tan 88^\circ$. Since $88^\circ = 90^\circ - 2^\circ$, we have $\tan 88^\circ = \tan(90^\circ - 2^\circ) = \frac{1}{\tan 2^\circ}$. Thus, $\tan 2^\circ \tan 88^\circ = \tan 2^\circ \times \frac{1}{\tan 2^\circ} = 1$.
• For the second pair: $\tan 3^\circ \tan 87^\circ$. Since $87^\circ = 90^\circ - 3^\circ$, we have $\tan 3^\circ \tan 87^\circ = \tan 3^\circ \times \frac{1}{\tan 3^\circ} = 1$.
• This pattern continues up to the pair $\tan 43^\circ \tan 47^\circ$. Since $47^\circ = 90^\circ - 43^\circ$, we have $\tan 43^\circ \tan 47^\circ = \tan 43^\circ \times \frac{1}{\tan 43^\circ} = 1$.

There are $(43 - 2) + 1 = 42$ such pairs, each evaluating to 1.

Evaluating the Middle Term

The expression also includes the term $\tan 45^\circ$. We know that:

Final Calculation

The original expression can be rewritten by substituting the value of each pair and the middle term:

The product consists of the values of all the pairs (which are all 1) multiplied by the value of $\tan 45^\circ$ (which is also 1).

Therefore, the value of the entire expression is 1.