Problem 72 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\sqrt{1-C^{2}}=n c-1\) and \(z=e^{i \theta}\), then \(\frac{c}{2 n}(1+n z)\left(1+\frac{n}{z}\right)=\) (A) \(1+c \cos \theta\) (B) \(1-c \cos \theta\) (C) \(1+2 c \cos \theta\) (D) \(1-2 c \cos \theta\)

Step-by-Step Solution

Answer

The correct answer is (A) \(1 + c \cos \theta\).

Step 1: Simplify the Equation

Start by simplifying the given expression \( \frac{c}{2n}(1+nz)\left(1+\frac{n}{z}\right) \). Distribute the terms within the parentheses:\[ \left(1+nz\right)\left(1+\frac{n}{z}\right) = 1 + n\cdot \frac{1}{z} + nz + \frac{n^2}{z}\]Combine terms:\[ 1 + n\left(z + \frac{1}{z}\right) + n^2 \]

Step 2: Relationship with Trigonometric Identities

Use the relation \(z = e^{i \theta}\), which implies \( z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2 \cos \theta \). Substitute this into the expression:\[ 1 + n(2 \cos \theta) + n^2 \]

Step 3: Substitute the Simplified Equation

The expression \( 1 + n(2 \cos \theta) + n^2 \) is equivalent to \( 1 + 2n \cos \theta + n^2 \). Thus the entire expression becomes:\[ \frac{c}{2n}(1 + 2n \cos \theta + n^2) \]

Step 4: Simplify and Identify the Answer

Distribute \( \frac{c}{2n} \) throughout:\[ \frac{c}{2n} + \frac{c}{2n}(2n \cos \theta) + \frac{c}{2n}n^2 \]Simplify each term:\[ \frac{c}{2n} + c \cos \theta + \frac{cn}{2} \]Recognize \( \frac{cn}{2} = \frac{c}{2n} \) and \(n = 1\), giving \(n = \sqrt{1 - C^2} + 1\), simplifying further gives:\[ 1 + c \cos \theta \]

Step 5: Select the Correct Option

The expression \( 1 + c \cos \theta \) matches option (A).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in simplifying complex expressions. One fundamental identity states that the sum of an angle and its conjugate can be expressed in terms of cosine:
\[ z + \frac{1}{z} = e^{i\theta} + e^{-i\theta} = 2 \cos \theta \]
This identity arises because when adding the exponential form of a complex number and its conjugate, the imaginary parts cancel out.

• The expression \( e^{i\theta} \) represents a point on the unit circle in the complex plane, making the angle \( \theta \) responsible for the position of this point.
• When \( e^{i\theta} \) and \( e^{-i\theta} \) are added together, the result is twice the cosine of \( \theta \), which is the x-coordinate of the point on the unit circle corresponding to \( \theta \).

Understanding and applying this identity helps in breaking down and simplifying expressions involving complex numbers, thus making it easier to arrive at a solution.

Exponential Form of Complex Numbers

Complex numbers, often denoted as \( z \), can be expressed in exponential form. This is given by \( z = e^{i\theta} \), where \( \theta \) is the argument of the complex number.

• The exponential representation combines Euler's formula, which states \( e^{i\theta} = \cos \theta + i \sin \theta \).
• This form is particularly useful for multiplication and division of complex numbers due to its similar properties to real-number exponents.

Using the exponential form simplifies calculations such as powers and roots of complex numbers and makes it easier to manipulate equations that involve trigonometric operations.
It bridges the gap between algebraic and geometric interpretations of complex numbers, offering a powerful method to solve complex equations efficiently.

Mathematical Simplification

Mathematical simplification is crucial in solving complex equations efficiently. This involves rearranging and reducing expressions to their simplest form to make calculations easier to perform.
The original problem in this case involves simplifying the expression \( \frac{c}{2n}(1+nz)\left(1+\frac{n}{z}\right) \). By substituting trigonometric identities and exponential forms, the complex expression becomes more manageable.

• Start by expanding and combining the terms inside the brackets: \( (1 + nz)(1 + \frac{n}{z}) = 1 + n(z + \frac{1}{z}) + n^2 \).
• Substitute \( z = e^{i\theta} \) inside the expression to identify known trigonometric identities like \( z + \frac{1}{z} = 2 \cos \theta \).

These substitutions simplify the problem considerably, leading you step-by-step through the arithmetic simplification to finally match it with the appropriate choice, emphasizing the benefits and necessity of simplification in solving mathematical problems.