Problem 257 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Find the value of } \left.\tan \frac{2 \pi}{5}-\tan \frac{\pi}{15}-\sqrt{3} \tan \frac{2 \pi}{5} \tan \frac{\pi}{15} \text { . \\{Ans. } \sqrt{3}\right\\} $$

Step-by-Step Solution

Answer

0

Step 1: Identify the formula

This problem requires the use of the tangent subtraction formula, which states that \( \tan (a-b)= \frac{(\tan a - \tan b)}{1+\tan a \cdot \tan b} \). Looking at the expression given, it's in the form of \( \tan a - \tan b - k \tan a \cdot \tan b \), which can be rearranged to obtain the above formula.

Step 2: Rearrange the expression

Rearrange the expression to the form of \( \tan(a-b) \) formula: \( \tan \frac{2 \pi}{5} - \tan \frac{\pi}{15} - \sqrt{3} \tan \frac{2 \pi}{5} \tan \frac{\pi}{15} = \tan \left( \frac{2 \pi}{5} - \frac{\pi}{15} \right) - \sqrt{3} \), when tan a = tan (2*pi/5) and tan b = tan (pi/15).

Step 3: Simplify the expression

Now simplify the inside of the tangent in the right side using the numerical values of \( a \) and \( b \): \( \tan \left( \frac{2 \pi}{5} - \frac{\pi}{15} \right) - \sqrt{3} = \tan \left( \frac{6 \pi}{15} \right) - \sqrt{3} = \tan \left( \frac{2 \pi}{5} \right) - \sqrt{3} \)

Step 4: Evaluate tan(2*pi/5)

Now, remember that tan(2*pi/5) = sqrt(3). Therefore the expression simplifies to \( \sqrt{3} - \sqrt{3} = 0 \)

Step 5: Conclusion

The value of the given expression is 0.

Key Concepts

Tangent Subtraction Formula

When solving trigonometric problems, one of the most handy tools is the tangent subtraction formula. This formula is useful when needing to find the tangent of the difference between two angles. Here’s how it works:

• The formula is expressed as: \[\tan(a-b)= \frac{(\tan a - \tan b)}{1+\tan a \cdot \tan b}\]
• This equation allows you to transform an expression involving the subtraction of two tangent terms into a single tangent operation.
• This is particularly advantageous in simplifying complex trigonometric expressions, as illustrated in our original exercise.

Understanding the tangent subtraction formula will make your trigonometric problem-solving much smoother. In our problem, identifying the expression \[\tan \left( \frac{2\pi}{5} - \frac{\pi}{15} \right)\]and relating it to this formula was the key initially. This transformation helps in pinpointing what each expression means in a trigonometric context, thus aiding in solving the problem efficiently.

Simplification in Trigonometry

Simplification is a vital skill in solving trigonometric problems. It involves rearranging and reducing expressions to their simplest form. By making expressions simpler, solving a problem becomes easier. Let's see how this applies in the tangent subtraction formula.

In the original exercise, the expression:

• \[\tan \frac{2 \pi}{5} - \tan \frac{\pi}{15} - \sqrt{3} \tan \frac{2 \pi}{5} \tan \frac{\pi}{15}\]
• was first recognized as being related to the tangent subtraction formula.
• The simplification began by identifying the expression as:\[\tan \left( \frac{2 \pi}{5} - \frac{\pi}{15} \right) - \sqrt{3}\]Significantly reducing the problem to a single tangent operation minus a known constant.
• Understanding this relationship allowed us to deduce that the simplified form, by substituting back in the predetermined values \(\sqrt{3}\), simplifies further to the final conclusion.

Practicing simplification techniques will not only make your solutions more readable but can uncover essential insights that lead you directly to the solution.

Mathematical Problem Solving

Mathematical problem solving involves a strategic approach to deciphering complex questions. Let's explore how this is applied in trigonometry using the problem at hand.

The process begins with understanding the problem fully. In our example:

• We needed to recognize the expression format and identify the valid trigonometric identities that can be utilized. Here, recognizing\(\tan \left( \frac{2\pi}{5} - \frac{\pi}{15} \right)\) was essential.
• Next, strategic rearrangement leads to correctly adapting the tangent subtraction formula, which simplifies the given terms into a workable expression.
• Further simplification of the terms allowed us to solve efficiently by rolling back through known values—here, it was crucial to remember predefined tangent values.
• Finally, when approaching any problem, checking the simplified form and solution accuracy ensures the correctness of each step.

Developing a structured method to solve such problems will enhance your efficiency and proficiency in trigonometry. This structured process exemplifies the elegance and logic inherent in mathematical problem solving.