Question

Use the identity \(\frac{1}{1-y}=\sum_{n=0}^{\infty} y^{n}\) to express the function as a geometric series in the indicated term. $$ \frac{1}{1+\sin ^{2} x} \text { in } \sin x $$

Short answer

Expressed as \( 1 - \sin^2 x + \sin^4 x - \sin^6 x + \ldots \) in terms of \( \sin x \).

Step-by-step solution

Identify the Original Formula

We are given the function \( \frac{1}{1+\sin^2 x} \). We need to express it using a geometric series formula in terms of \( \sin x \).

Rewrite the Denominator

Notice that \( \frac{1}{1+\sin^2 x} \) is close to the form \( \frac{1}{1-y} \) where \( y = \sin^2 x \). We aim to transform it for the series identity application. Rewriting it, \( \frac{1}{1-( - \sin^2 x)} \).

Substitute into the Series Formula

Substitute \( y = -\sin^2 x \) into the given series identity: \[ \frac{1}{1-y} = \sum_{n=0}^{\infty} y^{n} \]. Hence, \( \frac{1}{1+\sin^2 x} = \frac{1}{1-(-\sin^2 x)} = \sum_{n=0}^{\infty} (-\sin^2 x)^{n} \).

Expand the Geometric Series

Expand the series \( \sum_{n=0}^{\infty} (-\sin^2 x)^{n} \) to better understand its structure: \( 1 + (-\sin^2 x) + (-\sin^2 x)^2 + (-\sin^2 x)^3 + \ldots \).

Express in Terms of \( \sin x \)

Rewrite each term of the series in terms of \( \sin x \): \( 1 - \sin^2 x + (\sin^2 x)^2 - (\sin^2 x)^3 + \ldots \), ensuring each power is expressed using \( \sin x \).

Key concepts

Series Expansion

In mathematics, a series expansion is a powerful tool that allows us to express functions as sums of simpler terms. It's particularly helpful with functions that can be difficult to manage otherwise.
The core idea is to take a complex function and represent it as an infinite sum of terms, making it easier to handle and understand.
This approach is often used when dealing with functions in calculus, enabling easier computation for integration, differentiation, and evaluation at specific points.

• A common example is the geometric series expansion. It arises from the basic formula: \( \frac{1}{1-y}=\sum_{n=0}^{\infty} y^{n} \).
• This allows us to express functions with a format similar to \( \frac{1}{1-y} \) as a series, aiding in their simplification.

In solving the given exercise, this concept is applied to express the function \( \frac{1}{1+\sin^2 x} \) using a geometric series. By identifying the right substitution for \( y \), which is \( -\sin^2 x \) in this case, the formula \( \frac{1}{1-(-\sin^2 x)} = \sum_{n=0}^{\infty} (-\sin^2 x)^{n} \) enables the function to be expanded as a sum. This expansion forms the core solution in expressing the function with simpler polynomial terms.

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and various aspects of a triangle. Functions like sine and cosine are foundational in understanding wave behavior and circles.
The function that we work with in the exercise, \( \sin x \), is one of the most fundamental trigonometric functions.
It measures the ratio of the opposite side to the hypotenuse in a right-angled triangle.

• Trigonometric functions are periodic and exhibit cyclic behavior, making them essential in fields like physics and engineering for modeling waves and oscillations.
• The function \( \sin^2 x \) appears often in trigonometric identities and has useful properties in simplifying expressions involving trigonometric terms.

In the context of the original problem, we use \( \, \sin^2 x\, \) as a substitution within the geometric series expansion to simplify \( \, \frac{1}{1+\sin^2 x}\, \). By treating \( \, -\sin^2 x\, \) as \( \, y\, \), the function could be expressed in series, showcasing the flexibility of trigonometric functions in algebraic manipulation.

Infinite Series

An infinite series is a sum of an infinite sequence of terms. It extends indefinitely and lies at the heart of higher mathematics, offering a way to sum infinitely many terms to reach a finite result.
The infinite series is the backbone of many mathematical analyses, from approximating functions to deterministic modeling.

• This concept is crucial in calculus for defining convergent and divergent series which look closely at whether a series sums to a finite number.
• Series expansions, like the geometric series used here, rely on this concept to simplify expressions from an infinite perspective.

In our study of \( \frac{1}{1+\sin^2 x} \), the infinite series \( \sum_{n=0}^{\infty} (-\sin^2 x)^{n} \) provides a framework to express a complex rational function as an infinite sum or series.
Learning to work with infinite series develops a keen insight into understanding more about the convergence of sequences, laying foundational skills for further studies in mathematical theory and applications.