Problem 2 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that \(\operatorname{cosec} x+\operatorname{cosec} 2 x+\operatorname{cosec} 4 x+\ldots .\) to \(n\) terms \(=\cot \left(\frac{x}{2}\right)-\cot \left(2^{n-1} x\right)\)

Step-by-Step Solution

Answer

The mathematical proof of the given statement is true. The crucial strategy was to use mathematical induction: first, the statement was proven true for \( n = 1 \) (base case), then assuming it to be true for some arbitrary \( n = k \), it was proven for \( n = k + 1 \) (inductive step).

Step 1: Understand the relationship between \( \csc \) and \( \cot \)

The cosecant \( \csc x \) is the reciprocal of the sine function \( \sin x \). They have the relationship \( \csc x = \frac{1}{\sin x} \) and \( \cot x = \frac{\cos x}{\sin x} = \csc x \cdot \cos x \). Use these relationships to visualize the given problem.

Step 2: Use Mathematical Induction - Base Case

For mathematical induction, start with the base case. Here, the base case is when \( n = 1 \). Substituting \( n = 1 \) into the given expression gives \( \csc x = \cot(\frac{x}{2}) - \cot(2^{1-1}x) \) which simplifies to \( \csc x = \cot(\frac{x}{2}) - \cot x \). This proves true since \( \cot(\frac{x}{2}) = \csc x + \cot x \), which can be shown by using the half angle and double angle trigonometric identities.

Step 3: Use Mathematical Induction - Inductive Step

Assume that the equation is true for \( n = k \), i.e., \( S_k = \csc x + \csc 2x + \ldots + \csc 2^{k-1}x = \cot(\frac{x}{2}) - \cot 2^{k-1} x \). We need to prove that the equation is true for \( n = k + 1 \), i.e., \( S_{k+1} = \csc x + \csc 2x + \ldots + \csc 2^{k}x = \cot(\frac{x}{2}) - \cot 2^{k} x \).

Step 4: Prove the Inductive Step

Adding \( \csc 2^{k}x \) to both sides of the equation \( S_k \) gives \( S_k + \csc 2^{k}x = \cot(\frac{x}{2}) - \cot 2^{k} x + \csc 2^{k}x \). Since \( \csc 2y = \frac{2 \cot y}{1 - \cot^2 y} \), substituting \( y = 2^{k-1}x \) into this gives \( \csc 2^{k}x = \frac{2 \cot 2^{k-1}x}{1 - \cot^2 2^{k-1}x} \). Substituting this back into the equation with \( S_k \), we get \( S_{k+1} \), thus completing the inductive step.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for any value of the variable involved. They are useful for simplifying expressions and solving equations. One key identity involved in this exercise is

• \( \csc x = \frac{1}{\sin x} \)
• \( \cot x = \frac{\cos x}{\sin x} \)

By understanding these identities, we can transform complex trigonometric expressions into simpler forms. This simplification makes it easier to analyze and prove equations, such as the one in our original exercise. Recognizing these identities helps us see patterns and relationships between different trigonometric functions. As you progress through mathematical problems, knowing these identities will become increasingly valuable.

Cosecant Function

The cosecant function, denoted as \( \csc x \), is a trigonometric function that is the reciprocal of the sine function. It is defined as:\[ \csc x = \frac{1}{\sin x} \]This means it takes the value of the angle’s sine and flips it. Because it is the reciprocal of sine, \( \csc x \) shares the same fundamental period and undefined points (where \( \sin x = 0 \)) as the sine function.

• It becomes particularly useful in problems involving division of trigonometric functions, such as proving identities.
• Understanding how it interacts with other functions, such as \( \cos x \) in this problem, allows for more intricate problem-solving strategies.

Knowing how to manipulate \( \csc x \) within trigonometric identities is crucial to proving theoretical concepts found in trigonometry.

Cotangent Function

The cotangent function, represented as \( \cot x \), is another fundamental trigonometric function. It is expressed as the ratio of the cosine to sine of an angle:\[ \cot x = \frac{\cos x}{\sin x} \]Cotangent is essentially the reciprocal of tangent, reminiscent of how \( \csc x \) is the reciprocal of sine. This means:\[ \cot x = \frac{1}{\tan x} \]Understanding the cotangent function's role in trigonometric identities is key for solving and proving trigonometric equations.

• It helps in identifying \( \cot(\frac{x}{2}) - \cot(2^{n-1} x) \) within the steps of the proof.
• Using the properties of \( \cot x \) allows us to manipulate terms to support patterns found during mathematical induction applications.

The usage of \( \cot x \) is prevalent in both proving trigonometric identities and analyzing wave-like solutions.

Double Angle Formula

Double angle formulas are specific trigonometric identities related to the trigonometric functions of twice an angle. These formulas are immensely useful, particularly when trying to express complex trigonometric expressions in a simpler form.In this exercise, understanding the double angle identity for cosecant is crucial:

• \( \csc(2x) = \frac{2 \cot x}{1 - \cot^2 x} \)

This formula helps to transform terms like \( \csc 2^{k}x \) within the induction process used for proving the original expression.

• Double angle identities assist in transitioning from one trigonometric term to another, often making apparent connections that may not initially be visible.
• They allow for simplification, reducing larger expressions into more manageable components.

The application of the double angle formula is a testament to the elegance and power of trigonometric identities in mathematical proofs.