Problem 30 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of ' \(b\) ' such that the equation \(\frac{b \cos x}{2 \cos 2 x-1}=\frac{b+\sin x}{\left(\cos ^{2} x-3 \sin ^{2} x\right) \tan x}\) possess solutions, belongs to the set (A) \(\left(-\infty, \frac{1}{2}\right)\) (B) \(\left(\frac{1}{2}, \infty\right)\) (C) \((-\infty, \infty)\) (D) \(\left(-\infty, \frac{1}{2}\right) \cup(1, \infty)\)

Step-by-Step Solution

Answer

Option (D): \((-\infty, \frac{1}{2}) \cup (1, \infty)\)

Step 1: Simplify the Denominator

First, recognize that \( 2\cos 2x - 1 \) can be expressed using the double angle identity: \( \cos 2x = 2 \cos^2 x - 1 \). Therefore, \( 2\cos 2x - 1 = 4\cos^2 x - 2 \).

Step 2: Equate and Simplify

Now we equate the given equations: \( \frac{b \cos x}{4\cos^2 x - 2} = \frac{b + \sin x}{(\cos^2 x - 3 \sin^2 x) \tan x} \). Simplify the expression by multiplying across by the denominators: \( b \cos x (\cos^2 x - 3 \sin^2 x) \tan x = (b + \sin x)(4\cos^2 x - 2) \).

Step 3: Factor and Cancel Terms

Observe that the terms can be simplified further. Substitute \( \tan x = \frac{\sin x}{\cos x} \) and simplify. This might reveal common terms on both sides, allowing cancellation for simplification.

Step 4: Analyze the Conditions for the Solution

Consider the conditions under which the equation holds true, focusing on the properties of the trigonometric terms. Particularly, check for conditions where variables would lead to divisions by zero or undefined terms, especially since \( \tan x \) is involved.

Step 5: Determine Range for 'b'

Examine when the simplified version of the equation is valid. Evaluate against possible values of \( b \) so that no contradictions surface. Any scenario requiring \( b < \frac{1}{2} \) or \( b > 1 \) suggests option (D): \( \left(-\infty, \frac{1}{2}\right) \cup (1, \infty) \) as potential valid ranges.

Key Concepts

Double Angle Identity

Understanding the double angle identity is crucial when solving trigonometric equations. It allows us to simplify complex-looking terms into more manageable forms. In trigonometry, the double angle identity for cosine is given by \( \cos 2x = 2 \cos^2 x - 1 \). This identity transforms a double angle into a single angle, making calculations easier. For example, in our exercise, the term \( 2\cos 2x - 1 \) can be rewritten as \( 4\cos^2 x - 2 \). This helps simplify the denominator significantly. Recognizing these identities in equations is a key simplification technique. It reduces potential errors and aids in finding solutions faster. Always be on the lookout for angles or terms that can be expressed using double angle identities to streamline your work.

Simplification Techniques

When tackling a mathematical equation, simplification techniques are often your best friends. They transform complicated expressions into simpler forms. In the given problem, the use of simplification is evident. To simplify, first observe patterns or identities, such as factoring or substitution.Here are some key methods:

• Identify common trigonometric identities to transform expressions.
• Factorize complex terms whenever possible. This makes it easier to cancel out terms on both sides of the equation.
• Use substitutions to replace complex terms with simpler equivalents, like replacing \( \tan x \) with \( \frac{\sin x}{\cos x} \).

In the exercise provided, recognizing that \( \tan x = \frac{\sin x}{\cos x} \) allows for further simplification by canceling out terms. This, combined with factorization, exposes common factors and helps break down intricate parts of the equation into manageable pieces, paving the way for solving the equation efficiently.

Range of Values

One of the fundamental steps in solving equations is determining the range of values for which the equation holds true. This is crucial because mathematical expressions, particularly with trigonometric functions, might not be defined for certain values and can lead to contradictions.In this exercise, determining the acceptable range for \( b \) ensures solutions remain valid and consistent. Here’s how to approach it:

• Analyze conditions that could lead to division by zero, such as operations involving \( \tan x \), which becomes undefined at specific points.
• Check for any restrictions implied by the trigonometric identities used, ensuring all terms in the equation are defined across the desired range.
• Examine all possible scenarios that could restrict or validate specific ranges for the parameter \( b \).

Through careful analysis, it was found that \( b \) should not fall between \( \frac{1}{2} \) and 1 to avoid contradictions. Instead, it suggests that valid solutions can exist when \( b < \frac{1}{2} \) or \( b > 1 \), indicating a range of \( \left(-\infty, \frac{1}{2}\right) \cup (1, \infty) \). This comprehensive evaluation of value ranges helps ensure accuracy and completeness in deriving solutions.