Evaluating the Tangent Trigonometric Expression

The problem asks us to find the value of the expression: $ \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} + \sqrt{3}\tan \frac{2\pi}{7} \tan \frac{20\pi}{21} $

Using the Tangent Subtraction Identity

This expression resembles the tangent subtraction formula. The formula for the tangent of the difference between two angles $A$ and $B$ is given by: $ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} $

Let's identify the angles $A$ and $B$ from our expression. Let $A = \frac{20\pi}{21}$ and $B = \frac{2\pi}{7}$.

Calculating the Angle Difference

First, let's find the difference between the angles $A$ and $B$: $ A - B = \frac{20\pi}{21} - \frac{2\pi}{7} $ To subtract these fractions, we need a common denominator, which is 21: $ A - B = \frac{20\pi}{21} - \frac{2\pi \times 3}{7 \times 3} = \frac{20\pi}{21} - \frac{6\pi}{21} $ $ A - B = \frac{20\pi - 6\pi}{21} = \frac{14\pi}{21} $ Simplify the fraction: $ A - B = \frac{2\pi}{3} $

Applying the Formula

Now, let's substitute $A = \frac{20\pi}{21}$ and $B = \frac{2\pi}{7}$ into the tangent subtraction formula: $ \tan\left(\frac{2\pi}{3}\right) = \frac{\tan \frac{20\pi}{21} - \tan \frac{2\pi}{7}}{1 + \tan \frac{20\pi}{21} \tan \frac{2\pi}{7}} $

We know the value of $\tan\left(\frac{2\pi}{3}\right)$. The angle $\frac{2\pi}{3}$ is in the second quadrant, where tangent is negative. The reference angle is $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$. $ \tan\left(\frac{2\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3} $

So, we have: $ -\sqrt{3} = \frac{\tan \frac{20\pi}{21} - \tan \frac{2\pi}{7}}{1 + \tan \frac{20\pi}{21} \tan \frac{2\pi}{7}} $

Rearranging the Expression

To find the value of the given expression, let's rearrange the equation above. Multiply both sides by the denominator $(1 + \tan \frac{20\pi}{21} \tan \frac{2\pi}{7})$: $ -\sqrt{3} \left( 1 + \tan \frac{20\pi}{21} \tan \frac{2\pi}{7} \right) = \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} $ Distribute $-\sqrt{3}$ on the left side: $ -\sqrt{3} - \sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7} = \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} $ Now, move the term $-\sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7}$ to the right side of the equation: $ -\sqrt{3} = \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} + \sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7} $

This matches the expression we were asked to evaluate. Therefore, the value of the expression is $-\sqrt{3}$.