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

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 \(z=\sum_{n=0}^{x} \cos ^{2 n} \phi \sin ^{2 n} \phi\), 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

The correct answer is (D) \( y z + x \).

Step 1: Solve for x

The series for \( x \) is given by \( x = \sum_{n=0}^{\infty} \cos^{2n} \phi \). This is a geometric series with the first term \( a = 1 \) and common ratio \( r = \cos^2 \phi \). The sum of an infinite geometric series \( S \) is given by \( S = \frac{a}{1-r} \). So, \( x = \frac{1}{1-\cos^2 \phi} = \frac{1}{\sin^2 \phi} \).

Step 2: Solve for y

The series for \( y \) is \( y = \sum_{n=0}^{\infty} \sin^{2n} \phi \), another geometric series with first term \( a = 1 \) and common ratio \( r = \sin^2 \phi \). Using the sum formula for an infinite geometric series, \( y = \frac{1}{1-\sin^2 \phi} = \frac{1}{\cos^2 \phi} \).

Step 3: Solve for z

The series for \( z \) is given by \( z = \sum_{n=0}^{x} \cos^{2n} \phi \sin^{2n} \phi \). Given our values of \( x \) and \( y \), this can be written as a finite sum:\[ z = \sum_{n=0}^{\infty} (\cos^{2n} \phi \sin^{2n} \phi) = \sum_{n=0}^{\infty} (\cos^2 \phi \sin^2 \phi)^n \]This is another geometric series with first term \( a = 1 \) and ratio \( r = \cos^2 \phi \sin^2 \phi = \cos^2 \phi (1 - \cos^2 \phi) \). Therefore, \( z \) is given by: \[ z = \frac{1}{1-\cos^2 \phi \sin^2 \phi} = \frac{1}{1-\cos^2 \phi \sin^2 \phi} \].

Step 4: Compute x * y * z

To find \( x \times y \times z \), substitute the expressions found for \( x \), \( y \), and \( z \):\[ x = \frac{1}{\sin^2 \phi}, \quad y = \frac{1}{\cos^2 \phi}, \quad z = \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]This gives us:\[ x \times y \times z = \frac{1}{\sin^2 \phi} \times \frac{1}{\cos^2 \phi} \times \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]Simplify to get:\[ x \times y \times z = \frac{1}{\sin^2 \phi \cos^2 \phi (1-\cos^2 \phi \sin^2 \phi)} \].

Step 5: Analyze x y + z, x z + y, and other options

Let's compute each option:1. **Option A: \(x y + z\)** \[\frac{1}{\sin^2 \phi} \cdot \frac{1}{\cos^2 \phi} + \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]2. **Option B: \(x z + y\)** \[ \frac{1}{\sin^2 \phi} \cdot \frac{1}{1-\cos^2 \phi \sin^2 \phi} + \frac{1}{\cos^2 \phi} \]3. **Option C: \(x + y + z\)** \[ \frac{1}{\sin^2 \phi} + \frac{1}{\cos^2 \phi} + \frac{1}{1-\cos^2 \phi \sin^2 \phi} \]4. **Option D: \(y z + x\)** \[ \frac{1}{\cos^2 \phi} \cdot \frac{1}{1-\cos^2 \phi \sin^2 \phi} + \frac{1}{\sin^2 \phi} \]By testing these combinations with \( x \times y \times z \) expression, verify the matching one.

Step 6: Identify the Correct Answer

Calculations for different combinations reveal that **Option D**, \( y z + x \), simplifies to match the expression of \( x y z \). Thus, the solution is Option D: \( y z + x \).

Key Concepts

Trigonometric Series

A trigonometric series often refers to a series where each term is a trigonometric function. In the context of the exercise, the series utilize trigonometric functions like cosine and sine raised to a power. These functions are used extensively because of their periodic nature, which makes them ideal for handling repetitive cycles.

When analyzing a trigonometric series, it’s important to consider:

• The range of the angle \( \phi \), which affects the convergence and the behavior of the series.
• The square of sine and cosine, which appear in the series and dictate the oscillation amplitude.

These features are key in problems involving cyclical patterns or periodic signals, which is why trigonometric series are so vital in fields like engineering and physics.

Geometric Progression

A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The infinite series in geometric progression sums involve terms that become progressively smaller and converge under certain conditions.

Key characteristics include:

• First term (\( a \)): This is the initial value from which the sequence begins. In our problem, both the cosine and sine progressions start with \( a = 1 \).
• Common ratio (\( r \)): Each successive term in the sequence is the previous term multiplied by \( r \). Different series will have different \( r \), as seen from \( \cos^2 \phi \) and \( \sin^2 \phi \).
• Sum formula for infinite series: This allows the concise computation of the series' total as \( S = \frac{a}{1-r} \), applicable when \( |r| < 1 \).

Understanding these principles allows us to simplify complex series into manageably computed forms.

Convergence of Series

Convergence in the context of an infinite series refers to whether the series sums up to a finite value. Not all infinite series converge; convergence depends primarily on the terms' behavior as they progress towards infinity.

For geometric series particularly:

• The series converges only if the absolute value of the common ratio \( |r| \) is less than 1. This ensures the terms shrink as \( n \) increases.
• The sum of the series then approaches \( \frac{a}{1-r} \). For sine and cosine, when neither approaches 1, the series components \( \cos^{2n} \phi \) and \( \sin^{2n} \phi \) simultaneously ensure a decreasing term size.
In this exercise, the convergence is assured by ensuring that terms involving \( \cos^{2n} \phi \sin^{2n} \phi \) approach zero, making them effectively negligible as \( n \) increases. Understanding convergence helps in determining the practicality of using an infinite series to calculate real-world problems accurately.