Understanding the Trigonometric Expression

The problem asks us to find the value of the following trigonometric expression:

$ \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} + \sqrt{3}\tan \frac{2\pi}{7} \tan \frac{20\pi}{21} $

To solve this, we can utilize a key trigonometric identity involving the tangent function.

Applying the Tangent Subtraction Identity

Recall the tangent subtraction identity:

$ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} $

We can rearrange this formula to express $\tan A - \tan B$:

$ \tan A - \tan B = \tan(A - B)(1 + \tan A \tan B) $

Analyzing the Angles

Let's identify the angles in our expression:

• Angle $A = \frac{20\pi}{21}$
• Angle $B = \frac{2\pi}{7}$

To use the tangent subtraction formula effectively, we first find the difference between these angles:

$ A - B = \frac{20\pi}{21} - \frac{2\pi}{7} $

To subtract the fractions, we find 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} $

Simplifying the fraction:

$ A - B = \frac{2\pi}{3} $

Calculating the Tangent of the Angle Difference

Now, we find the tangent of this difference:

$ \tan(A - B) = \tan\left(\frac{2\pi}{3}\right) $

The value of $\tan\left(\frac{2\pi}{3}\right)$ is known:

$ \tan\left(\frac{2\pi}{3}\right) = -\sqrt{3} $

Connecting the Identity to the Given Expression

Using the rearranged tangent subtraction formula with $A = \frac{20\pi}{21}$ and $B = \frac{2\pi}{7}$:

$ \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} = \tan\left(\frac{2\pi}{3}\right) \left(1 + \tan \frac{20\pi}{21} \tan \frac{2\pi}{7}\right) $

Substitute the value of $\tan\left(\frac{2\pi}{3}\right)$:

$ \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) $

Distribute the $-\sqrt{3}$:

$ \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} = -\sqrt{3} - \sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7} $

Evaluating the Original Expression

The original expression we need to evaluate is:

$ E = \tan \frac{20\pi}{21} - \tan \frac{2\pi}{7} + \sqrt{3}\tan \frac{2\pi}{7} \tan \frac{20\pi}{21} $

Substitute the expression we found for $\tan \frac{20\pi}{21} - \tan \frac{2\pi}{7}$ into the equation for $E$:

$ E = \left(-\sqrt{3} - \sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7}\right) + \sqrt{3}\tan \frac{2\pi}{7} \tan \frac{20\pi}{21} $

Now, simplify the expression:

$ E = -\sqrt{3} - \sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7} + \sqrt{3} \tan \frac{20\pi}{21} \tan \frac{2\pi}{7} $

The last two terms cancel each other out:

$ E = -\sqrt{3} $

Final Answer

The value of the given trigonometric expression is $-\sqrt{3}$.