Problem 14 - A Complete Resource Book in Mathematics for JEE Main

Question

The maximum value of \(\left(\cos \alpha_{1}\right)\left(\cos \alpha_{2}\right) \cdot \dot{\pi} .(\cos a n)\) under the restrictions \(0 \leq \alpha_{1}, \alpha_{2}, \ldots, \alpha_{n} \leq \frac{\pi}{2}\) and (cot \(\left.\alpha_{1}\right)\left(\cot \alpha_{2}\right) \ldots\left(\cot \alpha_{n}\right)=1\) is (A) \(\frac{1}{2^{n / 2}}\) (B) \(\frac{1}{2^{n}}\) (C) \(\frac{1}{2 n}\) (D) 1

Step-by-Step Solution

Answer

The maximum value is \( \frac{1}{2^{n/2}} \).

Step 1: Understand the Problem

We need to find the maximum value of \( P = \cos \alpha_1 \cdot \cos \alpha_2 \cdot \ldots \cdot \cos \alpha_n \) given that \( \cot \alpha_1 \cdot \cot \alpha_2 \cdot \ldots \cdot \cot \alpha_n = 1 \). This constraint implies that the product of the cotangents is 1.

Step 2: Use the Trigonometric Identity

Recall the identity \( \cot \alpha = \frac{\cos \alpha}{\sin \alpha} \). Therefore, the constraint \( \cot \alpha_1 \cdot \cot \alpha_2 \cdot \ldots \cdot \cot \alpha_n = 1 \) can be rewritten as:\[\frac{\cos \alpha_1}{\sin \alpha_1} \cdot \frac{\cos \alpha_2}{\sin \alpha_2} \cdot \ldots \cdot \frac{\cos \alpha_n}{\sin \alpha_n} = 1\]This simplifies to:\[\frac{\cos \alpha_1 \cdot \cos \alpha_2 \cdot \ldots \cdot \cos \alpha_n}{\sin \alpha_1 \cdot \sin \alpha_2 \cdot \ldots \cdot \sin \alpha_n} = 1\]

Step 3: Simplify the Equation

From the constraint after simplification, we have:\[\cos \alpha_1 \cdot \cos \alpha_2 \cdot \ldots \cdot \cos \alpha_n = \sin \alpha_1 \cdot \sin \alpha_2 \cdot \ldots \cdot \sin \alpha_n\]Let us denote this product as \( P \). Thus, \( P = \sin \alpha_1 \cdot \sin \alpha_2 \cdot \ldots \cdot \sin \alpha_n \).

Step 4: Apply AM-GM Inequality

The Arithmetic Mean-Geometric Mean Inequality (AM-GM) states that for any non-negative numbers, the geometric mean is less than or equal to the arithmetic mean. Applying AM-GM to the cosines:Let \( x_i = \cos \alpha_i \) and \( y_i = \sin \alpha_i \), then\[P = x_1 \cdot x_2 \cdot \ldots \cdot x_n = y_1 \cdot y_2 \cdot \ldots \cdot y_n\]AM-GM for \( \cos^2 \alpha_i + \sin^2 \alpha_i = 1 \) leads to the maximum of \( x_i = \frac{1}{\sqrt{2}} \).

Step 5: Determine the Maximum Value

For each \( \cos \alpha_i \), the maximum value satisfying the constraint is \( \cos \alpha_i = \sin \alpha_i = \frac{1}{\sqrt{2}} \). This makes sense when \( \alpha_i = \frac{\pi}{4} \) for all \( i \). Hence:\[P_{max} = \left(\frac{1}{\sqrt{2}}\right)^n = \frac{1}{2^{n/2}}\]

Step 6: Choose the Correct Option

The maximum value of the product \( P \) is \( \frac{1}{2^{n/2}} \). Therefore, the correct option is (A) \( \frac{1}{2^{n / 2}} \).

Key Concepts

Arithmetic Mean-Geometric Mean Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality is a vital tool in mathematics. It's a concept that helps to compare the arithmetic mean and geometric mean of non-negative numbers. According to this inequality, for any set of non-negative numbers, the arithmetic mean is always greater than or equal to the geometric mean. Mathematically, for any non-negative numbers \( a_1, a_2, ..., a_n \):

• Arithmetic Mean (AM) = \( \frac{a_1 + a_2 + ... + a_n}{n} \)
• Geometric Mean (GM) = \( \sqrt[n]{a_1 \cdot a_2 \cdots a_n} \)

We have the inequality: \[\frac{a_1 + a_2 + ... + a_n}{n} \geq \sqrt[n]{a_1 \cdot a_2 \cdots a_n}\] The AM-GM inequality is especially useful when dealing with maximization or minimization problems, as it provides an upper or lower boundary for a product of numbers when their sum is fixed. This inequality played a crucial role in the solution, as it helped determine the maximum possible value for the product of cosines.

Cosine Function

The cosine function is one of the primary trigonometric functions, usually abbreviated as \( \cos \). It's a fundamental part of trigonometry, and it relates the adjacent side of a right-angled triangle to its hypotenuse. In trigonometric identities and formulas, it's written as:\[\cos(\alpha) = \frac{\text{Adjacent side}}{\text{Hypotenuse}}\]Where \( \alpha \) is the angle in question. Understanding the cosine function in terms of the unit circle can also be helpful: - The cosine of an angle is the x-coordinate of the corresponding point on the unit circle.- It is periodic, repeating its values in regular intervals of \( 2\pi \).In the exercise, the cosine values \( \cos \alpha_i \) were analyzed to find the maximum product given specific constraints. These constraints played a role in determining the values of \( \alpha_i \) that would yield the largest product of cosines.

Cotangent Function

The cotangent function, denoted as \( \cot \), is the reciprocal of the tangent function in trigonometry. It can be expressed in terms of sine and cosine as follows:\[\cot(\alpha) = \frac{1}{\tan(\alpha)} = \frac{\cos(\alpha)}{\sin(\alpha)}\]The cotangent function is defined for all angles except where the sine function is zero, which happens at integer multiples of \( \pi \). It's useful in many trigonometric expressions and identities.In this exercise, we utilized the product \[\cot \alpha_1 \cdot \cot \alpha_2 \cdots \cot \alpha_n = 1\]to derive necessary conditions on \( \alpha_i \) that help simplify and solve the problem. The constraint was rephrased using cotangents to establish a balance between products of sines and cosines needed for the desired maximization.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold for all values of the involved variables. They are fundamental tools in simplifying trigonometric expressions, solving trigonometric equations, and solving geometric problems through trigonometry.Some basic identities include:

• Pythagorean identities: \(\sin^2(\alpha) + \cos^2(\alpha) = 1\)
• Reciprocal identities: \(\cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)}\)

These identities are invaluable, as seen in our exercise. The cotangent identity was vital to transition from the product constraints to a solvable form using known identities. Similarly, understanding relationships like \(\sin^2(\alpha) + \cos^2(\alpha) = 1\) helped in determining the angle values that maximize the product of cosine terms. Learning and applying these identities can greatly enhance a student's ability to tackle a wide range of mathematical problems.