Which of the following equations are identities?
In mathematics, an identity is an equation that holds true for all values of the variables involved. Identities are fundamental in algebra and trigonometry, as they provide a set of true statements that can be used to simplify expressions and solve equations. In this article, we will examine a set of equations and determine which of them are identities.
1. \( a^2 + b^2 = c^2 \)
2. \( (x + y)^2 = x^2 + 2xy + y^2 \)
3. \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
4. \( \frac{a + b}{c} = \frac{a}{c} + \frac{b}{c} \)
5. \( x^3 – y^3 = (x – y)(x^2 + xy + y^2) \)
Let’s analyze each equation to determine if it is an identity.
1. \( a^2 + b^2 = c^2 \): This equation is known as the Pythagorean theorem and is true for all right-angled triangles. Therefore, it is an identity.
2. \( (x + y)^2 = x^2 + 2xy + y^2 \): This equation is the binomial expansion of the square of a sum. It holds true for all real numbers \( x \) and \( y \), making it an identity.
3. \( \sin^2(\theta) + \cos^2(\theta) = 1 \): This equation is a fundamental trigonometric identity, known as the Pythagorean identity. It holds true for all real values of \( \theta \), so it is an identity.
4. \( \frac{a + b}{c} = \frac{a}{c} + \frac{b}{c} \): This equation is an algebraic identity that holds true for all real numbers \( a \), \( b \), and \( c \), as long as \( c \) is not equal to zero. Therefore, it is an identity.
5. \( x^3 – y^3 = (x – y)(x^2 + xy + y^2) \): This equation is known as the difference of cubes formula. It holds true for all real numbers \( x \) and \( y \), making it an identity.
In conclusion, all the given equations are identities. They are fundamental in mathematics and can be used to simplify expressions and solve equations involving the variables involved. Understanding these identities is crucial for students of algebra and trigonometry to excel in their mathematical studies.
