Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(a_{1}=\left(\tan \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{2}=\left(\tan \frac{\pi}{8}\right)^{\cos \frac{\pi}{8}}\), \(a_{3}=\left(\cot \frac{\pi}{8}\right)^{\tan \frac{\pi}{8}}, a_{4}=\left(\cot \frac{\pi}{8}\right)^{\cos \frac{\pi}{8}}\) Then, (A) \(a_{4}>a_{3}>a_{2}>a_{1}\) (B) \(a_{3}>a_{4}>a_{2}>a_{1}\) (C) \(a_{4}>a_{3}>a_{1}>a_{2}\) (D) \(a_{3}>a_{1}>a_{2}>a_{4}\)

Step-by-Step Solution

Answer

The correct order is (B) \(a_{3}>a_{4}>a_{2}>a_{1}\).

Step 1: Evaluate trigonometric ratios

Calculate the values of \( \tan \frac{\pi}{8} \) and \( \cot \frac{\pi}{8} \). We know that \( \tan \frac{\pi}{8} = \sqrt{2} - 1 \) and \( \cot \frac{\pi}{8} = \sqrt{2} + 1 \).

Step 2: Evaluate each term

Compute the values of \( a_1, a_2, a_3, \) and \( a_4 \):1. \( a_1=(\sqrt{2}-1)^{\sqrt{2} - 1} \)2. \( a_2=(\sqrt{2}-1)^{\cos \frac{\pi}{8}} \)3. \( a_3=(\sqrt{2}+1)^{\sqrt{2} - 1} \)4. \( a_4=(\sqrt{2}+1)^{\cos \frac{\pi}{8}} \)

Step 3: Compare powers

Since \( \cos \frac{\pi}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2} \) and is approximately 0.92, and \( \sqrt{2} - 1 \) is approximately 0.41, compare the powers of these terms.

Step 4: Analyze base properties

Since \( \sqrt{2}-1 \) is less than 1 and \( \sqrt{2}+1 \) is greater than 1, apply the property that decreasing the power (exponent less than 1) results in larger values for bases greater than 1, and smaller values for bases less than 1.

Step 5: Final ordering

For \( \sqrt{2}-1 \), powers decreasing result in smaller values, so \( a_2 > a_1 \). For \( \sqrt{2}+1 \), powers decreasing result in larger values, so \( a_3 > a_4 \). Arrange as \( a_3 > a_4 > a_2 > a_1 \).

Key Concepts

Tan and Cot Identities

In trigonometry, tangent (tan) and cotangent (cot) are fundamental trigonometric functions. These functions help in relating the angles of a triangle to its sides. Tangent, or \( \tan \theta \), is defined as the ratio of the opposite side to the adjacent side in a right triangle. Similarly, cotangent, \( \cot \theta \), is the ratio of the adjacent side to the opposite side. These identities are useful in solving equations involving trigonometric expressions.
To evaluate expressions involving \( \tan \frac{\pi}{8} \) and \( \cot \frac{\pi}{8} \), we can use known values:

• \( \tan \frac{\pi}{8} = \sqrt{2} - 1 \)
• \( \cot \frac{\pi}{8} = \sqrt{2} + 1 \)

These values are derived from the mathematical relationships in trigonometric identities and allow us to compute problems involving these angles.

Inequalities

Inequalities are a mathematical way to express the relative sizes or order of two numbers or expressions. They show whether one number is less than, greater than, or approximately equal to another. In our problem, inequalities allow us to compare the expressions \( a_1, a_2, a_3, \) and \( a_4 \) based on their calculated values.

• If \( x > y \), then the inequality \( x \) is greater than \( y \).
• If \( x < y \), then the inequality expresses \( x \) is less than \( y \).

When evaluating these trigonometric-based expressions:

• For bases less than 1, decreasing exponents lead to higher values because the fraction between 0 and 1 gets larger as the exponent decreases.
• Conversely, for bases greater than 1, decreasing exponents cause the whole number to decrease.

This understanding helps in rearranging terms by their magnitude and solving the inequality correctly. Comparing expressions like \( \sqrt{2}-1 \) and \( \sqrt{2}+1 \) with their respective exponents solidifies the ordering of these terms.

Powers and Exponents

Powers and exponents are crucial mathematical concepts, indicating repeated multiplication of a base number. The expression \( a^n \) is known as 'a' raised to the power of 'n'. This is crucial when working with expressions in problems such as the given trigonometric problems.

• If the base \( a \) is less than 1, increasing the exponent makes \( a^n \) a smaller fraction.
• If the base \( a \) is more than 1, increasing the exponent makes \( a^n \) grow larger.

In our exercise, constants like \( \cos \frac{\pi}{8} \) (\[ \frac{\sqrt{2 + \sqrt{2}}}{2} \] yielding approximately 0.92) and \( \sqrt{2} - 1 \) (approximately 0.41) are used as exponents. Understanding how these exponents interact with the base values helps determine the final ordering. By analyzing how each base interacts with its exponent—considering whether it's less or greater than 1—allows the correct comparison and arrangement of \( a_1, a_2, a_3, \) and \( a_4 \).