Problem 29 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\) are in A.P. whose common difference is \(\alpha\), then the value of \(\sin \alpha\left[\sec x_{1} \sec x_{2}+\sec x_{2} \sec \right.\) \(\left.x_{3}+\cdots+\sec x_{n-1} \sec x_{n}\right]\) is equal to (A) \(\frac{\sin n \alpha}{\cos x_{1} \cos x_{n}}\) (B) \(\frac{\sin (n-1) \alpha}{\cos x_{1} \cos x_{n}}\) (C) \(\frac{\sin (n+1) \alpha}{\cos x_{1} \cos x_{n}}\) (D) none of these

Step-by-Step Solution

Answer

The answer is option (A): \(\frac{\sin n \alpha}{\cos x_1 \cos x_{n}}\).

Step 1: Understand the Sequence and Write Terms

Given that \(x_1, x_2, x_3, \ldots, x_n\) are in an arithmetic progression (A.P.) with common difference \(\alpha\). This means \(x_2 = x_1 + \alpha\), \(x_3 = x_1 + 2\alpha\), ..., \(x_n = x_1 + (n-1)\alpha\).

Step 2: Analyze the Expression Inside the Sum

We need to find the value of \(\sin \alpha \left[ \sec x_1 \sec x_2 + \sec x_2 \sec x_3 + \, \ldots , + \sec x_{n-1} \sec x_n \right]\). The pattern involves terms like \(\sec x_i \sec x_{i+1}\). These can be written as \( \frac{1}{\cos x_i \cos x_{i+1}}\).

Step 3: Simplify Using Known Trigonometric Identities

Recall the identity for the sum \(\sum_{i=1}^{n-1} \cos(x_i) \cos(x_{i+1}) = \frac{1}{2} [\cos(x_1 - x_2) + \cos(x_1 + x_2)]\). However, we deal with secants, invert the terms: \(\sec x_i \sec x_{i+1} = \frac{2}{\cos(x_i - x_{i+1}) - \cos(x_i + x_{i+1})}\).

Step 4: Use Angle Difference and Sum Formulas

Given that \(x_{i+1} = x_i + \alpha\), substitute to find \(\cos(x_i - x_{i+1}) = \cos(-\alpha) = \cos \alpha\), and \(\cos(x_i + x_{i+1})\) will form a term considering the given sequence in A.P. Convert this into a trigonometric sum identity.

Step 5: Apply the Formula for Sum of Finite Series

Convert the entire sequence using known trigonometric formulas. The summation over all these will form a telescopic pattern, reducing into a single formula \( \frac{\sin n \alpha}{\cos x_1 \cos x_{n}}\), using known identities for telescopic sum simplification.

Step 6: Identify the Correct Answer

Comparing the final formula determined from our reduction \( \frac{\sin n \alpha}{\cos x_1 \cos x_{n}}\) with the provided options, it matches option (A). Thus, the value of the expression evaluates to option (A).

Key Concepts

arithmetic progression

Arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This constant is called the "common difference." For example, if we have a sequence: 1, 3, 5, 7, ..., the common difference here is 2. In the context of the exercise, if we are given that terms such as \(x_1, x_2, x_3, \ldots, x_n\) are in A.P. with a common difference \(\alpha\), it implies:

• \(x_2 = x_1 + \alpha\)
• \(x_3 = x_1 + 2\alpha\)
• ... and so on, until \(x_n = x_1 + (n-1)\alpha\)

If you are given the first term and the common difference, you can generate the entire sequence. This systematic property is crucial for solving the problem and knowing how sequences follow a predictable pattern.

common difference

The common difference \(\alpha\) is the key ingredient that defines an arithmetic progression. It's the fixed amount added to each term to get to the next one. Suppose we know the first term \(x_1\) and the common difference \(\alpha\), we can write any term in the sequence as:

• Nth term, \(x_n = x_1 + (n-1)\alpha\)

Understanding the common difference is critical because it allows us to find relationships and patterns within the sequence. In trigonometric calculations, especially when dealing with sequences like in the given exercise, knowing \(\alpha\) helps to approach the expression simplification based on the trigonometric identities relative to these terms.

telescopic series

A telescopic series is a sequence of terms designed such that most terms cancel in pairs, simplifying the computation of the total sum. In mathematical expressions, a telescopic series exploits these cancellations to make complex sums easier to solve. To construct a telescopic series in the given exercise, think about the trigonometric product terms such as \(\sec x_i \sec x_{i+1}\). These can be rewritten by using different identities to reveal the telescopic pattern. By cleverly applying algebraic and trigonometric manipulations:

• Each term contributes mainly two parts: \(\cos(x_i - x_{i+1})\) and \(\cos(x_i + x_{i+1})\).
• The sum simplifies greatly through these identity transformations, giving us a reduced elegant expression involving fewer terms.

This conceptual approach turns a potentially cumbersome problem into a manageable calculation. In this solution, all the internal terms cancel each other out, resulting in the simplified form \(\frac{\sin n \alpha}{\cos x_1 \cos x_n}\).

angle sum formulas

Angle sum formulas are identities in trigonometry that provide the sine, cosine, or tangent of sums and differences of angles. They are crucial for breaking down complex trigonometric expressions. For this exercise, consider the trigonometric identity for cosine of angle sums: \[\cos(x + y) = \cos x \cos y - \sin x \sin y\]This formula allows decomposition of composite angles into functions of the individual angles, which is invaluable for evaluating terms like \(\cos(x_i + x_{i+1})\). Similarly, when handling expressions like \(\sec x_i \), it transforms to \(\frac{1}{\cos x_i}\), making use of angle sum formulas to express terms like \(\sec(x_1 + x_2)\) becomes straightforward.In the context of solving the exercise, applying these formulas aids in breaking down complex summations involving secant terms into manageable pieces, facilitating simplification and solution finding. These tools are powerful, allowing expressions to be rearranged into forms where telescoping and summing up become possible.