Evaluating the Trigonometric Expression

The problem asks us to find the value of the expression \( \rm{tan} \dfrac{5 \pi}{12} - \rm{tan} \dfrac{\pi}{12} - \sqrt 3 \rm{tan} \dfrac{5 \pi}{12}. \rm{tan} \dfrac{ \pi}{12} \). This expression involves tangent functions of specific angles.

Understanding the Tangent Subtraction Formula

A key trigonometric identity that is useful here is the tangent subtraction formula:

\[ \tan(A - B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B} \]

We can rearrange this formula to get:

\[ \tan(A - B) (1 + \tan A \tan B) = \tan A - \tan B \]

Expanding the left side gives:

\[ \tan(A - B) + \tan(A - B) \tan A \tan B = \tan A - \tan B \]

Rearranging this equation, we get a form that looks more similar to the given expression:

\[ \tan A - \tan B - \tan(A - B) \tan A \tan B = \tan(A - B) \]

Applying the Formula to the Given Angles

In the given expression, we have \(\tan \dfrac{5 \pi}{12}\) and \(\tan \dfrac{\pi}{12}\). Let's consider \(A = \dfrac{5 \pi}{12}\) and \(B = \dfrac{\pi}{12}\).

First, let's find the difference between these two angles:

\[ A - B = \dfrac{5 \pi}{12} - \dfrac{\pi}{12} = \dfrac{5 \pi - \pi}{12} = \dfrac{4 \pi}{12} = \dfrac{\pi}{3} \]

Now, let's find the tangent of this difference:

\[ \tan(A - B) = \tan \left(\dfrac{\pi}{3}\right) \]

The value of \(\tan(\pi/3)\) is a standard trigonometric value:

\[ \tan \left(\dfrac{\pi}{3}\right) = \sqrt{3} \]

So, \(\tan(A - B) = \sqrt{3}\).

Connecting the Formula to the Expression

Using our rearranged formula: \[ \tan A - \tan B - \tan(A - B) \tan A \tan B = \tan(A - B) \] Substitute \(A = \dfrac{5 \pi}{12}\), \(B = \dfrac{\pi}{12}\), and \(\tan(A - B) = \sqrt{3}\):

\[ \tan \dfrac{5 \pi}{12} - \tan \dfrac{\pi}{12} - \sqrt{3} \tan \dfrac{5 \pi}{12} \tan \dfrac{ \pi}{12} = \sqrt{3} \]

The expression we are asked to evaluate is exactly the left-hand side of this equation:

\[ \rm{tan} \dfrac{5 \pi}{12} - \rm{tan} \dfrac{\pi}{12} - \sqrt 3 \rm{tan} \dfrac{5 \pi}{12}. \rm{tan} \dfrac{ \pi}{12} \]

Therefore, the value of the given expression is equal to the right-hand side of the equation, which is \(\sqrt{3}\).

Final Calculation Steps

1. Identify the angles: \(A = \dfrac{5 \pi}{12}\), \(B = \dfrac{\pi}{12}\).
2. Calculate the difference: \(A - B = \dfrac{5 \pi}{12} - \dfrac{\pi}{12} = \dfrac{4 \pi}{12} = \dfrac{\pi}{3}\).
3. Calculate \(\tan(A - B)\): \(\tan \left(\dfrac{\pi}{3}\right) = \sqrt{3}\).
4. Recall the rearranged tangent subtraction formula: \( \tan A - \tan B - \tan(A - B) \tan A \tan B = \tan(A - B) \).
5. Substitute the values: \( \tan \dfrac{5 \pi}{12} - \tan \dfrac{\pi}{12} - \sqrt{3} \tan \dfrac{5 \pi}{12} \tan \dfrac{ \pi}{12} = \sqrt{3} \).

The value of the expression is \(\sqrt{3}\).

Revision Table: Key Concepts in Trigonometry

Additional Information on Trigonometric Identities

Trigonometric identities are equations that are true for all possible values of the variables involved. They are fundamental tools in simplifying expressions, solving equations, and proving other relationships in trigonometry.

• The tangent addition formula, \( \tan(A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B} \), is closely related to the subtraction formula. You can derive the subtraction formula from the addition formula by substituting \(B\) with \(-B\), keeping in mind that \(\tan(-B) = -\tan B\).
• Angles like \(\dfrac{\pi}{12}\) (\(15^\circ\)) and \(\dfrac{5 \pi}{12}\) (\(75^\circ\)) can be expressed as sums or differences of standard angles, e.g., \(\dfrac{\pi}{12} = \dfrac{\pi}{4} - \dfrac{\pi}{6}\) or \(\dfrac{\pi}{3} - \dfrac{\pi}{4}\), and \(\dfrac{5 \pi}{12} = \dfrac{\pi}{4} + \dfrac{\pi}{6}\). You could evaluate \(\tan \dfrac{5 \pi}{12}\) and \(\tan \dfrac{\pi}{12}\) separately using the sum/difference formulas and then substitute into the original expression, but the method using the rearranged formula is more direct in this specific case.
• Knowing the standard trigonometric values for angles like \(0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, \dfrac{\pi}{2}, \pi\), etc., is crucial for solving many trigonometry problems.