(a+1/a)^2+(a-1/a)^2

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Jul 03, 2024

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(a+1/a)^2 + (a-1/a)^2: A Mathematical Exploration

In this article, we will delve into the fascinating world of algebra and explore the expression (a+1/a)^2 + (a-1/a)^2. We will simplify this expression, identify its properties, and discuss its significance in various mathematical contexts.

Simplifying the Expression

To simplify the expression, we can start by expanding the squares:

(a+1/a)^2 = (a^2 + 2a + 1)/a^2

and

(a-1/a)^2 = (a^2 - 2a + 1)/a^2

Now, let's add the two expressions:

(a^2 + 2a + 1)/a^2 + (a^2 - 2a + 1)/a^2

Combining like terms, we get:

(2a^2 + 2)/a^2

Simplifying further, we arrive at:

2 + 2/a^2

Properties of the Expression

The simplified expression 2 + 2/a^2 has several interesting properties:

• Symmetry: The expression is symmetric with respect to the replacement of a with -a.
• Positivity: The expression is always positive, regardless of the value of a.
• Asymptotic Behavior: As a approaches infinity, the expression approaches 2.

Applications and Significance

The expression (a+1/a)^2 + (a-1/a)^2 appears in various areas of mathematics, including:

• Trigonometry: This expression is related to the sum of squares of sine and cosine functions.
• Algebraic Identities: The expression is used to prove various algebraic identities, such as the one mentioned above.
• Analysis: The expression is used in analysis to study the properties of functions and their limits.

In conclusion, the expression (a+1/a)^2 + (a-1/a)^2 is a fascinating mathematical entity with rich properties and diverse applications. Its simplicity and elegance make it a valuable tool for mathematicians and scientists alike.