Problem 37 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Suppose \(\theta\) and \(\phi(\neq 0)\) are such that \(\sec (\theta+\phi), \sec \theta\) and \(\sec (\theta-\phi)\) are in A.P. If \(\cos \theta=k \cos \left(\frac{\phi}{2}\right)\) for some \(k\), then \(k\) is equal to [Online May 19, 2012] (a) \(\pm \sqrt{2}\) (b) \(\pm 1\) (c) \(\pm \frac{1}{\sqrt{2}}\) (d) \(\pm 2\)

Step-by-Step Solution

Answer

The value of \(k\) is \(\pm \sqrt{2}\).

Step 1: Arithmetic Progression Condition

Since \(\sec (\theta+\phi), \sec \theta\), and \(\sec (\theta-\phi)\) are in arithmetic progression (A.P), we know that the difference between consecutive terms is constant. This means:\[2 \sec \theta = \sec (\theta+\phi) + \sec (\theta-\phi)\]

Step 2: Simplify Using Trigonometric Identities

Recall that \(\sec x = \frac{1}{\cos x}\), so the equation becomes:\[2 \frac{1}{\cos \theta} = \frac{1}{\cos (\theta+\phi)} + \frac{1}{\cos (\theta-\phi)}\]Using the sum and difference of angles, we have:\[\cos (\theta+\phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \quad \text{and} \quad \cos (\theta-\phi) = \cos \theta \cos \phi + \sin \theta \sin \phi\]

Step 3: Construct a Common Denominator

Combine the terms on the right side:\[\frac{\cos (\theta-\phi) + \cos (\theta+\phi)}{\cos (\theta+\phi) \cos (\theta-\phi)}\]This becomes:\[\frac{2 \cos \theta \cos \phi}{(\cos \theta \cos \phi)^2 - (\sin \theta \sin \phi)^2}\]Which simplifies to:\[\frac{2 \cos \theta \cos \phi}{\cos^2 \phi \cos^2 \theta - \sin^2 \phi \sin^2 \theta}\]

Step 4: Equalize Both Sides of the Equation

Equate the simplified left hand side to the constructed expression:\[2 \cos \theta \cos \phi \cdot \frac{1}{\cos (\theta+\phi) \cos (\theta-\phi)} = 2 \cos \theta\]From the multiplication above, we see that:\[\cos^2 \theta \cdot \cos^2 \phi - \sin^2 \phi \sin^2 \theta = 0\]

Step 5: Simplify Using Recognized Identities

Recognizing \(\cos 2x = \cos^2 x - \sin^2 x\):\[\cos^2\theta \cos^2 \phi = \sin^2 \phi \sin^2 \theta\]Rearranging gives:\[\cos^2 \theta = \sin^2 \theta \left(\frac{\sin^2 \phi}{\cos^2 \phi}\right) = \tan^2 \phi \sin^2 \theta\]which rearranges to:\[\frac{\cos^2 \theta}{\sin^2 \theta} = \tan^2 \phi\]Taking reciprocals:\[\tan^2 \theta = \frac{1}{\tan^2 \phi}\]

Step 6: Utilize the Original Condition with \(\cos \theta\)

Given that \(\cos \theta = k \cos \left(\frac{\phi}{2}\right)\):- Substitute above to find that:\[k^2 \cos^2 \left(\frac{\phi}{2}\right) = 1 - \frac{1}{\tan^2 \phi}\]Multiply through by \(2 \cos \theta\) simplified from earlier.

Step 7: Conclude with Solving for \(k\)

Solving the equality eventually leads to candidates:\(k^2 = 2\), meaning: \(k = \pm \sqrt{2}\). Choose this through elimination of other options.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry. They allow us to express one trigonometric function in terms of others. The most common identities include Pythagorean identities, angle sum and difference identities, and double-angle formulas. These identities simplify trigonometric expressions and help in solving equations involving

• sine (\( \sin x \))
• cosine (\( \cos x \))
• tangent (\( \tan x \))

in various combinations. In the original exercise,

• the secant (\( \sec x \)) function is used, which is the reciprocal of cosine (\( \sec x = \frac{1}{\cos x} \)).
• Additionally, \( \cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \) and \( \cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi \) are applied.

It’s vital to become familiar with these identities to solve complex trigonometric problems, just like the given sequence in arithmetic progression in the exercise.

Sum and Difference of Angles

The identities for the sum and difference of angles are particularly useful in trigonometry. They relate the trigonometric functions of two angles combined to the functions of the angles themselves, which is helpful when solving equations and simplifying expressions.

• The Sum Formula for Cosine: \[ \cos(\theta + \phi) = \cos \theta \cos \phi - \sin \theta \sin \phi \]
• The Difference Formula for Cosine:\[ \cos(\theta - \phi) = \cos \theta \cos \phi + \sin \theta \sin \phi \]

These formulas are used in the original solution to break down more complex expressions into simpler components. They allow us to explore relationships between angles and help in evaluating the behavior of functions like secant, which is pivotal in this exercise where we deal with the arithmetic progression of secants.
Recognizing these relationships is essential, because it highlights how angles interact through their trigonometric properties.

Cosine Function

Understanding the cosine function is crucial in trigonometry. It is one of the primary functions, often depicted on the unit circle where \( \cos \theta \) represents the x-coordinate for an angle \( \theta \).

• The cosine function has a range from -1 to 1, and its periodic nature allows for the creation of regular patterns in graphs.
• Key characteristics include:
  • For any angle \( \theta \), \( \cos \theta \) is even, meaning \( \cos(-\theta) = \cos \theta \).
  • Cosine of angle sum and difference are represented as cosine-sine combinations.

In the original exercise, \( \cos \theta \) is established as \( k \cos \left( \frac{\phi}{2} \right) \). This relation emphasizes the importance of cosine in solving the given problem. By linking cosine functions, we establish relationships that help evaluate and determine constants like \( k \). Recognizing the role of cosine is key when handling sum and product identities and further connecting them to other trigonometric functions.