Which of the following are identities? Check all that apply.
In mathematics, identities are fundamental equations that hold true for all values of the variables involved. They are not just equations that are true for specific values, but rather universal truths that apply across the board. In this article, we will explore some common identities and determine which ones are indeed identities.
1. a^2 – b^2 = (a + b)(a – b)
This is a well-known algebraic identity known as the difference of squares. It states that the square of a number minus the square of another number is equal to the product of the sum and difference of those two numbers. This identity holds true for all real numbers a and b, making it a valid identity.
2. (a + b)^2 = a^2 + 2ab + b^2
This is the binomial expansion of the square of a sum. It states that the square of the sum of two numbers is equal to the sum of the squares of those numbers, plus twice the product of the numbers. This identity is also valid for all real numbers a and b, and thus qualifies as an identity.
3. sin^2(x) + cos^2(x) = 1
This is a fundamental trigonometric identity known as the Pythagorean identity. It states that the square of the sine of an angle plus the square of the cosine of the same angle is always equal to 1. This identity holds true for all real numbers x, making it an identity.
4. 1 + 1 = 2
This is a simple arithmetic identity that states that the sum of two ones is equal to two. While this may seem trivial, it is still an identity because it holds true for all real numbers. Therefore, it is a valid identity.
5. (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3
This is the binomial expansion of the cube of a sum. It states that the cube of the sum of two numbers is equal to the sum of the cubes of those numbers, plus three times the product of the first number squared and the second number, plus three times the product of the first number and the second number squared, and finally, the cube of the second number. This identity is valid for all real numbers x and y, making it an identity.
In conclusion, all the listed identities (1, 2, 3, 4, and 5) are indeed identities. They are fundamental equations that hold true for all values of the variables involved, making them essential tools in various mathematical fields.
