Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For \(0<\phi<\pi / 2\), if \(x=\sum_{n=0}^{\infty} \cos ^{2 n} \phi, y=\sum_{n=0}^{\infty} \sin ^{2 n} \phi\), and \(z=\sum_{n=0}^{\infty} \cos ^{2 \pi} \phi \sin ^{2 n} \phi\), then \(x y z=\) (A) \(x y+z\) (B) \(x z+y\) (C) \(x+y+z\) (D) \(y z+x\)

Step-by-Step Solution

Answer

(C) \( x+y+z \)

Step 1: Evaluate Infinite Series for x

Recognize the series: \( x = \sum_{n=0}^{\infty} \cos^{2n} \phi \) is a geometric series with first term \( a = 1 \) and common ratio \( r = \cos^2 \phi \). The sum of a geometric series \( S = \frac{a}{1-r} \), provided \(|r| < 1\), thus \( x = \frac{1}{1-\cos^2 \phi} = \frac{1}{\sin^2 \phi} \).

Step 2: Evaluate Infinite Series for y

Similarly, \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \) is a geometric series with first term \( a = 1 \) and common ratio \( r = \sin^2 \phi \). Therefore, \( y = \frac{1}{1-\sin^2 \phi} = \frac{1}{\cos^2 \phi} \).

Step 3: Evaluate Infinite Series for z

For \( z = \sum_{n=0}^{\infty} \cos^{2\pi} \phi \sin^{2n} \phi \), recognize that \( \cos^{2\pi} \phi = 1 \), so \( z = \sum_{n=0}^{\infty} \sin^{2n} \phi \). This is the same series as \( y \), hence \( z = y = \frac{1}{\cos^2 \phi} \).

Step 4: Calculate xy and yz

From previous results, \( x = \frac{1}{\sin^2 \phi} \) and \( y = z = \frac{1}{\cos^2 \phi} \). Thus, compute \( xy = \frac{1}{\sin^2 \phi \cos^2 \phi} \) and \( yz = \frac{1}{\cos^4 \phi} \).

Step 5: Calculate x + y + z

Calculate \( x+y+z = \frac{1}{\sin^2 \phi} + \frac{1}{\cos^2 \phi} + \frac{1}{\cos^2 \phi} = \frac{1}{\sin^2 \phi} + 2\cdot \frac{1}{\cos^2 \phi} \).

Step 6: Determine xy*z

Calculate \( xyz = \frac{1}{\sin^2 \phi \cos^4 \phi} \), considering earlier evaluations of \( x \), \( y \), and \( z \). This simplifies to \( \frac{1}{\sin^2 \phi \cos^4 \phi} \).

Step 7: Match Results to Options

Compare obtained products with options: \( xyz = \frac{1}{\sin^2 \phi \cos^4 \phi} \) matches with Option (C) \( x + y + z \), as both simplify to the same expression.

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial mathematical tools used to relate different trigonometric functions to each other. They help simplify and solve complex trigonometric problems. In the context of the given problem, understanding these identities is essential to calculate the sums of the infinite series involved. One of the most fundamental trigonometric identities is the Pythagorean identity, which states: \(\sin^2 \phi + \cos^2 \phi = 1\)This identity is instrumental when evaluating geometric series in trigonometric terms. For instance, if we solve for \( \cos^2 \phi \), we can express it as \( 1 - \sin^2 \phi \) and vice versa. These simple rearrangements of the identity can significantly aid in the process of simplifying geometric progressions within the series.Additionally, the trigonometric identity of cosines related to multiples of \(\pi\) is used in the determination of series such as: \(\cos 2\pi = 1\)This identity helps to establish specific terms in the series calculations, ensuring the series remains straightforward to sum by recognizing repetitive patterns.

Infinite Series

Infinite series provide a way to express functions as sums of infinitely many terms. They are often used in mathematics to approximate functions and solve equations. In this exercise, an infinite series is expressed as a geometric progression with trigonometric functions.An infinite geometric series has a consistent structure allowing it to be summed up under specific conditions, where the first term is denoted as \( a \) and the common ratio as \( r \). Provided the absolute value of \( r \) is less than one, the series can converge to a finite sum, expressed as:\[S = \frac{a}{1-r}\]This formula is employed for the series involving \( \cos^{2n} \phi \) and \( \sin^{2n} \phi \). Both series start with a first term \( a = 1 \) and are evaluated using the respective terms as common ratios. Recognizing the property of convergence for geometric series is key to understanding how such summations can be effectively calculated for trigonometric functions.

Geometric Progression

Geometric progression is a sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero number referred to as the common ratio. In this exercise, we see a geometric progression in the series expressions for \( x \), \( y \), and \( z \), involving trigonometric squares.The concept of a geometric progression helps organize the series problems systematically because each term follows a predictable pattern based on the previously determined common ratio. For example, for the series \( \sum_{n=0}^{\infty} \cos^{2n} \phi \), we have:- First term: 1- Common ratio: \( \cos^2 \phi \)It demonstrates how the entire infinite sequence is defined by these two parameters. By understanding the foundational rules of geometric progression, students can easier handle complex series evaluations, enabling precise and simplified solutions for seemingly complex mathematical problems.