Problem 39 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f_{n}(\theta)=\tan \frac{\theta}{2}(1+\sec \theta)(1+\sec 2 \theta)(1+\sec 4 \theta) \ldots .\) \(\left(1+\sec 2^{n} \theta\right)\), then (A) \(f_{2}\left(\frac{\pi}{16}\right)=1\) (B) \(f_{3}\left(\frac{\pi}{32}\right)=1\) (C) \(f_{4}\left(\frac{\pi}{64}\right)=1\) (D) \(f_{5}\left(\frac{\pi}{128}\right)=1\)

Step-by-Step Solution

Answer

(B) is correct: \(f_{3}\left(\frac{\pi}{32}\right) = 1\).

Step 1: Simplify the Function Expression

The function \( f_{n}(\theta) = \tan \frac{\theta}{2} (1+\sec \theta)(1+\sec 2\theta)(1+\sec 4\theta) \cdots (1+\sec 2^n \theta) \) can be simplified by considering that \( 1 + \sec \theta = \frac{2}{\cos \theta} = \frac{1 + \tan^2(\theta/2)}{(\cos\theta/2)^2} \). This suggests a telescoping pattern when calculating the entire product, potentially simplifying as all cosines cancel each other out.

Step 2: Verify Telescoping Argument

The expression simplifies to a telescoping series if analyzed step by step: \( \tan \frac{\theta}{2} = \frac{\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}} \), and when you include \((1+\sec 2\theta)(1+\sec 4\theta)\cdots\), each step can be paired with subsequent terms to simplify to 1.

Step 3: Apply for Specific Values

Apply the formula to specific values. For \( f_{2}\left(\frac{\pi}{16}\right) \), \( f_{3}\left(\frac{\pi}{32}\right) \), etc. calculate \( \tan\frac{\theta}{2} \) and each successive \( 1 + \sec 2^k \theta \) until \( 2^k = 2^n \). Use angle identities where necessary to simplify.

Step 4: Direct Calculation for \( f_3\left(\frac{\pi}{32}\right) \)

Calculate \( f_3\left(\frac{\pi}{32}\right) \). Start with \( \tan\left(\frac{\pi}{64}\right) \, (1 + \sec\left(\frac{\pi}{32}\right)) \, (1 + \sec\left(\frac{\pi}{16}\right)) \, (1 + \sec\left(\frac{\pi}{8}\right)) \), recognizing telescoping and the identity simplifications. We see the function reduces to 1 for \(\theta = \frac{\pi}{32}\), using our earlier simplifications.

Key Concepts

Telescoping Series

A telescoping series is a sequence or a related series where most terms cancel each other out, leaving just a few terms behind. The magic of such a series lies in its structure, where sequentially paired terms simplify the entire sum or product. With these series, solving complex expressions becomes easier.

• Breakdown: Each new term you introduce in a telescoping pattern "cancels out" part of a previous term.
• Application: In our exercise, the function's product simplifies remarkably, revealing itself as a telescoping series, showing how one might reduce a seemingly complex function to a simpler form.

Telescoping series are useful because they often change intimidating, lengthy calculations into concise expressions. You start with something long and, through the magic of cancellation, end with something short!

Secant Function

The secant function, written as \( \sec(\theta) \), is one of the six fundamental trigonometric functions. Defined as the reciprocal of the cosine function, \( \sec(\theta) = \frac{1}{\cos(\theta)} \), it is crucial in many trigonometric identities and expressions.

• Properties: \( \sec(\theta) \) is undefined whenever \( \cos(\theta) = 0 \). It features vertical asymptotes at these points in its graph.
• Usage: Often shows up in expressions needing simplification through identities such as \( 1 + \sec(\theta) = \frac{2}{\cos(\theta)} \).
• In our problem: \( 1 + \sec(\theta) \) contributes to the telescoping pattern, allowing replacement by other trigonometric identities for easier simplification.

Understanding the secant function helps in transforming complicated trigonometric expressions into more manageable forms and is essential in the world of mathematics, especially calculus and analytic geometry.

Tangent Function

The tangent function, \( \tan(\theta) \), is another vital trigonometric function, defined as the ratio of the sine and cosine functions: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \). It possesses unique properties and significance in trigonometry.

• Behavior: \( \tan(\theta) \) has periods of undefined values where \( \cos(\theta) = 0 \), featuring vertical asymptotes that repeat at intervals of \( \pi \).
• Special Identity: The half-angle identity \( \tan\frac{\theta}{2} = \frac{\sin\frac{\theta}{2}}{\cos\frac{\theta}{2}} \), is often employed in simplifying and solving trigonometric equations.
• Our exercise: \( \tan\frac{\theta}{2} \) initiates the telescoping pattern, which, when paired with terms involving \( \sec(\theta) \), swiftly reduces down to a simpler expression.

Mastering the tangent function enhances your ability to navigate trigonometric calculations and provides insights into angle measure transformations and periodic behaviors in mathematical functions.