Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

The sum of the series \(\sin \theta \cdot \sec 3 \theta+\sin 3 \theta \cdot \sec 3^{2} \theta+\) \(\sin 3^{2} \theta \sec 3^{3} \theta+\ldots\) up to \(n\) terms is (A) \(\frac{1}{2}\left(\tan 3^{n} \theta-\tan \theta\right)\) (B) \(\left(\tan 3^{n} \theta-\tan \theta\right)\) (C) \(\tan 3^{n} \theta-\tan 3^{n-1} \theta\) (D) none of these

Step-by-Step Solution

Answer

Option (B): \( \tan 3^{n} \theta - \tan \theta \).

Step 1: Understand the series

We observe that the series is given by \( \sin \theta \cdot \sec 3 \theta + \sin 3\theta \cdot \sec 3^{2} \theta + \sin 3^{2} \theta \cdot \sec 3^{3} \theta + \ldots \) up to \( n \) terms. This series can be written in a general form as \( \sin (3^{k-1} \theta) \sec (3^k \theta) \) for \( k = 1, 2, ..., n \).

Step 2: Simplify each term

Observe that \( \sin(3^{k-1} \theta) \sec(3^{k} \theta) = \tan(3^{k-1} \theta) \). Each term in the series can thus be expressed as \( \tan(3^{k-1} \theta) \).

Step 3: Analyze the sum

We need to find the sum \( \sum_{k=1}^{n} \tan(3^{k-1} \theta) \). Notice that this is a telescoping series.

Step 4: Identify the telescoping pattern

The series has the form \( \tan(\theta), \tan(3\theta), \tan(3^2\theta), \ldots, \tan(3^{n-1}\theta) \). The telescoping nature is visible because the sum can simplify due to cancellation.

Step 5: Calculate the sum

The sum of a telescoping series typically simplifies between the first and the last terms. Therefore, the total sum is \( \tan(3^n \theta) - \tan(\theta) \).

Step 6: Conclude the solution

Compare this result to the options given. The expression matches option (B) \( \left( \tan 3^{n} \theta - \tan \theta \right) \).

Key Concepts

Telescoping Series

When facing a sequence of terms like the one in this exercise, it's helpful to recognize patterns. The concept of a telescoping series applies when parts of each term cancel out with parts of succeeding or preceding terms. In its simplest form, a telescoping series is one where most terms will cancel each other out, typically leaving only the first and the last term.

• In our exercise, each term is expressed as \( \tan(3^{k-1} \theta) \), forming a sequence from \( \tan(\theta) \) to \( \tan(3^{n-1} \theta) \).
• Such a sequence is described as telescoping because if you sum these terms, many intermediate values will cancel out.
• The beauty and utility here are the simplification of complex summation problems into straightforward results.

Ultimately, the sum simplifies to the difference between the final and initial terms, which is \( \tan(3^n \theta) - \tan(\theta) \). By understanding this principle, solving problems like these becomes merely a matter of setting up the series and applying the telescoping rule.

Trigonometric Identities

Understanding trigonometric identities is crucial in simplifying expressions and solving trigonometric summations. For the series in the exercise, recognizing that \( \sin(x) \cdot \sec(x) = \tan(x) \) is vital. This identity stems from the definition of secant and tangent:

• The secant of an angle, \( \sec x \), is the reciprocal of its cosine, i.e., \( \sec x = \frac{1}{\cos x} \).
• The tangent of an angle, \( \tan x \), is the ratio of its sine to its cosine, i.e., \( \tan x = \frac{\sin x}{\cos x} \).
• Thus, \( \sin(x) \cdot \sec(x) = \frac{\sin x}{1/\cos x} = \tan x \).

By applying these identities, we transformed the initial series into a telescoping one. Remember, a solid grasp of these identities simplifies many mathematical problems, making complex series not only manageable but often down-right simple.

Series Sum Calculation

Calculating a series sum effectively relies on understanding its structure and type. Given that the series in question telescopes, the summation process simplifies significantly.

• First, you recognize and express each term in a suitable form, here as \( \tan(3^{k-1} \theta) \).
• Next, determine the series' pattern or behavior, in this case, telescoping.
• With telescoping series, calculate the sum by focusing on the initial and final terms after cancellation, leading to \( \tan(3^n \theta) - \tan(\theta) \).
• Compare the result against given options to verify correct solutions.

By accurately identifying the series type, you can reduce complex tasks into quick and efficient calculations. This method ensures precision and saves time, particularly in examinations or timed settings.