Problem 67 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a_{1}, a_{2}, \ldots, a_{n}\) are in A.P. with common difference \(d \neq 0\), then sum of the series \(\sin d\left[\sec a_{1} \sec a_{2}+\sec a_{2}\right.\) \(\left.\sec a_{3}+\ldots+\sec a_{n-1} \sec a_{n}\right]\) is (A) \(\tan a_{n}-\tan a_{1}\) (B) \(\cot a_{n}-\cot a_{1}\) (C) \(\sec a_{n}-\sec a_{1}\) (D) \(\operatorname{cosec} a_{n}-\operatorname{cosec} a_{1}\)

Step-by-Step Solution

Answer

The sum is \(\tan a_n - \tan a_1\), so the answer is (A).

Step 1: Understand the Series in A.P.

In an arithmetic progression (A.P.), each term is obtained by adding a fixed number, known as the common difference \(d\), to the previous term. Thus, we have \(a_{k} = a_{1} + (k-1)\cdot d\). In this problem, \(a_1, a_2, \ldots, a_n\) are terms of an A.P.

Step 2: Analyze the Sum Expression

The expression we need to analyze is \(\sin d \left[ \sec a_1 \sec a_2 + \sec a_2 \sec a_3 + \ldots + \sec a_{n-1} \sec a_n \right]\). This expression finds the sum of products of secants of consecutive terms in the A.P., each multiplied by \(\sin d\).

Step 3: Use Trigonometric Identities

Recalling the identity \(\sec x = \frac{1}{\cos x}\) and the fact that \(\sin d(\sec x \sec y) = \frac{1}{\cos x \cos y} \cdot \sin d\), we simplify the expression where needed.

Step 4: Swap and Simplify

We know from trigonometric identities that \(\sin d \cdot \sec x \sec y = \sin d \cdot \frac{1}{\cos x \cos y} = \tan x - \tan y\) when the terms are select properly. Recognizing the pattern here is key to summing an apparent telescopic series.

Step 5: Identify the Telescopic Series

The elements form a telescopic series, which means terms cancel each other conveniently. Specifically, \((\tan a_2 - \tan a_1) + (\tan a_3 - \tan a_2) + \ldots + (\tan a_n - \tan a_{n-1})\) simplifies down to \(\tan a_n - \tan a_1\).

Step 6: Compare with Options

After simplifying, compare results with the given options. We observe that the sum simplifies to \(\tan a_n - \tan a_1\), which matches option (A).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable where both sides of the identity are defined. These identities are extremely useful in simplifying complex expressions, especially in the realm of series and sequences.

One of the key identities used in this problem is the relationship between secant and cosine, stated as \( \sec x = \frac{1}{\cos x} \). This identity allows the transformation of secant terms into reciprocal cosine terms, which can then be manipulated further using other trigonometric identities.

In the exercise, another crucial identity applied is the simplification \( \sin d \cdot \sec x \cdot \sec y = \tan x - \tan y \) under the right conditions. This identity is fundamental in recognizing the pattern of the telescopic series, which turns potentially complicated expressions into simple differences between tangent terms. Understanding these identities is essential for working efficiently with trigonometric series.

Telescopic Series

A telescopic series is a type of series where many terms cancel out sequentially with previous or subsequent terms, leading to a vastly simpler expression after summation. This technique relies on recognizing patterns that simplify the sequence of operations, often resulting in a very manageable final term.

In this exercise, the series involves secant terms of consecutive arithmetic progression terms, configured such that they fit a telescopic pattern. When written out, the terms \((\tan a_2 - \tan a_1) + (\tan a_3 - \tan a_2) + \ldots + (\tan a_n - \tan a_{n-1})\) perfectly cancel, leaving only \(\tan a_n - \tan a_1\).

Recognizing a telescopic series can drastically reduce the time needed to find a solution by focusing attention on only the first and last terms. This method is particularly useful when dealing with long sequences and is a valuable tool in mathematical problem-solving.

Common Difference

The concept of common difference is central to arithmetic progressions (A.P.). It represents the fixed amount added to each term to get the next term in the sequence. This "step" between terms is what distinguishes arithmetic progressions from other types of sequences, and it remains constant throughout the series.

Mathematically, if \(a_1, a_2, \ldots, a_n\) is an arithmetic progression, each term can be expressed as \(a_k = a_1 + (k-1) \cdot d\), where \(d\) is the common difference. This formula facilitates the calculation of any term in the series if the first term and common difference are known.

In this exercise, the common difference \(d\) is also woven into the trigonometric expression \( \sin d \) that multiplies the sum of secant products. Understanding the role of the common difference is crucial to unraveling expressions and discovering underlying series patterns, such as the telescopic series identified within this problem.