Problem 31 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Prove that, \((2 \cos \theta-1)(2 \cos 2 \theta-1)\left(2 \cos 2^{2} \theta-1\right)\) \(\ldots\left(2 \cos 2^{n-1} \theta-1\right)=\left(\frac{2 \cos \left(2^{n} \theta\right)+1}{2 \cos \theta+1}\right)\)

Step-by-Step Solution

Answer

The given identity is valid for all positive integers n. The crucial part in the proof was to utilize the concept of mathematical induction, combined with careful application of trigonometric identities and equations.

Step 1: Base Case

Start by checking the base case where \( n = 1 \). For \( n = 1 \), the left hand side of the equation is \( (2\cos\theta - 1) \) and the right hand side is \( (2\cos2\theta + 1)/(2\cos\theta + 1) \). Simplifying the right hand side we get \( (2\cos\theta - 1) \), which matches the left side. Thus, the base case holds true.

Step 2: Inductive Hypothesis

Assume that the given identity is true for some positive integer \( k \), that is, \( (2\cos\theta - 1)(2\cos2\theta - 1)\ldots(2\cos2^{k-1}\theta -1) = (2\cos2^{k}\theta + 1) / (2\cos\theta + 1) \). This is our inductive hypothesis.

Step 3: Inductive Step

We prove the identity for \( n = k + 1 \), if it's true for \( n = k \). For \( n = k + 1 \), the left hand side can be expanded to \( (2\cos\theta - 1)(2\cos2\theta - 1)\ldots(2\cos2^{k}\theta -1)(2\cos2^{k}\theta -1) \). As per our inductive hypothesis, the first \( k \) terms give \( (2\cos2^k\theta + 1)/(2\cos\theta + 1) \). Then, multiplying the (k+1)th term gives \( ((2\cos2^{k+1}\theta + 1)(2\cos2^{k}\theta - 1)) / (2\cos\theta + 1) \). Simplifying this we get \( (2\cos2^{k+1}\theta + 1) / (2\cos\theta + 1) \), which is the expression for \( n = k + 1 \) on the right hand side, proving our induction.

Key Concepts

Induction in Trigonometry

Induction is a powerful technique often used to prove statements that are formulated for all natural numbers. In trigonometry, induction can help verify complex identities. The process involves two main steps: the base case and the inductive step.

First, you begin with the **base case**. This is where you show that the statement holds true for the initial value, often when \( n = 1 \). In our exercise, we check that the identity holds for \((2\cos\theta - 1)\), and indeed both sides of the equation match.

Next, the **inductive hypothesis** is assumed. You presume the identity is true for \( n = k \). It forms the crux of the induction as it bridges the known with the unknown.

The final piece is the **inductive step**. Here, you demonstrate that if the statement holds for \( n = k \), it must also hold for \( n = k + 1 \). By mastering this technique, you're essentially "building" the statement's truth for every integer incrementally. As you work through these steps, you're effectively using logical progression to unravel complex trigonometric identities.

Cosine Function

The cosine function is one of the core trigonometric functions, represented as \( \cos \theta \). It defines the x-coordinate of a point on the unit circle as the angle \( \theta \) is swept from the positive x-axis.

**Properties of Cosine**:

• It is an even function, meaning \( \cos(-\theta) = \cos(\theta) \).
• Its range is between -1 and 1.
• It has a periodicity of \( 2\pi \), meaning \( \cos(\theta + 2\pi) = \cos \theta \).
• It is related to other trigonometric functions, such as sine, through Pythagorean identities: \( \cos^2 \theta + \sin^2 \theta = 1 \).

In the given exercise, the cosine function is iteratively applied to powers of two, forming expressions like \( 2\cos2^k\theta \). These expressions exploit the recursive nature of the cosine function, leading to simplifications and derivations as evident in trigonometric identities.

Simplification of Expressions

Simplifying expressions involves reducing them to their most basic form. This plays a critical role in solving and proving trigonometric identities.

In this exercise, simplification occurs when applying identities and algebraic manipulation to break down complex expressions. For example, when dealing with terms like \((2\cos2^k\theta - 1)\), understanding how each term interacts leads to a successful simplification.

To achieve simplification, several tools are essential:

• **Factorization**: Breaking down terms can reveal hidden patterns, simplifying the expression significantly.
• **Trigonometric Identities**: Use known identities like angle-sum and difference formulas or double angle formulas to transform expressions.
• **Algebraic manipulation**: Combine like terms, solve fractions, and apply basic arithmetic to ease complexity.

Mastering simplification requires practice and a solid grasp of both algebra and trigonometric identities. As you simplify, look for opportunities to substitute back into known values or expressions, paving the way toward proving or solving complex trigonometric results.