Which Of The Following Equations Are Identities

Algebra thrives on relationships, and understanding those relationships is key to unlocking more complex mathematical concepts. Among these relationships, identities hold a special place. They are equations that are true for all values of the variables involved. Unlike regular equations that are only true for specific values, identities offer a universal truth within the mathematical landscape.

Navigating the world of algebraic equations often involves discerning whether a given equation is an identity. This distinction is crucial because identities allow for simplification, manipulation, and a deeper understanding of the underlying mathematical structure. Let's delve into the methods for verifying identities and explore numerous examples to solidify our comprehension.

What is an Identity?

An identity is an equation that remains true regardless of the value assigned to its variables. It's like a universally accepted rule in mathematics. For example, the equation x + x = 2x is an identity because it holds true no matter what number we substitute for x. On the other hand, x + 2 = 5 is not an identity because it is only true when x = 3.

The power of identities lies in their ability to transform expressions without changing their fundamental value. This is immensely useful for simplifying complex equations, solving problems more efficiently, and gaining a deeper insight into the relationships between different mathematical concepts.

Methods for Verifying Identities

Several methods can be used to determine if an equation is an identity. The most common include:

• Simplification: Manipulate one or both sides of the equation using algebraic rules until they are identical. If you can transform one side into the other, the equation is an identity.
• Substitution: Substitute various values for the variable(s) and check if the equation holds true for each value. While this doesn't prove an identity, it can provide strong evidence. If you find even one value that makes the equation false, you know it's not an identity.
• Expansion: Expand any products or powers in the equation. This can reveal hidden simplifications and make it easier to compare the two sides.
• Factoring: Factor expressions on either side of the equation. This can help to identify common factors and simplify the expression.
• Using Known Identities: Leverage established identities (like trigonometric identities, algebraic identities, etc.) to transform the equation into a recognizable form.

Let's now put these methods into practice with a variety of examples.

Examples of Determining Identities

We will now examine several equations and determine whether they represent identities using the methods described above.

Example 1: Is (a + b)² = a² + b² an identity?

• Method: Let's use simplification and expansion. We know the correct expansion of (a + b)² is:

  (a + b)² = (a + b)(a + b) = a² + 2ab + b²

  Comparing this to the right side of the original equation (a² + b²), we see they are not the same unless 2ab = 0. This only happens when a = 0 or b = 0. Since the original equation is not true for all values of a and b, it is not an identity.

• Substitution: Let's try a = 1 and b = 2:

  (1 + 2)² = 3² = 9

  1² + 2² = 1 + 4 = 5

  Since 9 ≠ 5, the equation is not an identity.

Example 2: Is x² - 1 = (x + 1)(x - 1) an identity?

• Method: Simplification/Expansion. We can expand the right side of the equation:

  (x + 1)(x - 1) = x² - x + x - 1 = x² - 1

  This is exactly the same as the left side of the equation. Therefore, the equation is an identity.

• Factoring: We can factor the left side of the equation using the difference of squares:

  x² - 1 = (x + 1)(x - 1)

  This directly matches the right side, confirming the equation is an identity.

Example 3: Is sin²(θ) + cos²(θ) = 1 an identity?

• Method: This is a fundamental trigonometric identity. It is true for all values of θ. Therefore, the equation is an identity. This is often referred to as the Pythagorean Identity in trigonometry.

Example 4: Is x + 5 = 8 an identity?

• Method: Substitution. This equation is only true when x = 3. For any other value of x, the equation is false. Therefore, it is not an identity.

Example 5: Is 2(y + 3) = 2y + 6 an identity?

• Method: Simplification/Expansion. Distribute the 2 on the left side:

  2(y + 3) = 2y + 6

  This is exactly the same as the right side of the equation. Therefore, the equation is an identity.

Example 6: Is |x| = x an identity?

• Method: Substitution. Let's try x = -2:

  |-2| = 2

  Since 2 ≠ -2, the equation is not an identity. The absolute value of a number is only equal to the number itself when the number is non-negative.

Example 7: Is (x + y)³ = x³ + y³ an identity?

• Method: Simplification/Expansion. We need to expand (x + y)³:

  (x + y)³ = (x + y)(x + y)(x + y) = (x² + 2xy + y²)(x + y) = x³ + 3x²y + 3xy² + y³

  This is not equal to x³ + y³ for all values of x and y. Therefore, the equation is not an identity.

• Substitution: Let x = 1 and y = 1:

  (1 + 1)³ = 2³ = 8

  1³ + 1³ = 1 + 1 = 2

  Since 8 ≠ 2, the equation is not an identity.

Example 8: Is cos(-θ) = cos(θ) an identity?

• Method: This is a known trigonometric identity. The cosine function is an even function, meaning cos(-θ) = cos(θ) for all values of θ. Therefore, the equation is an identity.

Example 9: Is tan(θ) = sin(θ)/cos(θ) an identity?

• Method: This is the definition of the tangent function in trigonometry. It holds true for all values of θ where cos(θ) ≠ 0 (because division by zero is undefined). Therefore, the equation is an identity.

Example 10: Is a² - b² = (a - b)(a + b) an identity?

• Method: Simplification/Expansion. Expanding the right side:

  (a - b)(a + b) = a² + ab - ab - b² = a² - b²

  This matches the left side. Therefore, the equation is an identity.

Example 11: Is log(xy) = log(x) + log(y) an identity?

• Method: This is a logarithmic identity. However, it's important to note that it only holds true when x > 0 and y > 0 because the logarithm of a non-positive number is undefined. Assuming x and y are positive, the equation is an identity.

Example 12: Is e^ln(x) = x an identity?

• Method: This is an identity related to the exponential and natural logarithm functions. It is true for all x > 0 because the natural logarithm is only defined for positive numbers. For x > 0, the equation is an identity.

Example 13: Is sin(2θ) = 2sin(θ) an identity?

• Method: This is a trigonometric identity, but the correct identity is sin(2θ) = 2sin(θ)cos(θ). Therefore, sin(2θ) = 2sin(θ) is not an identity.

• Substitution: Let θ = π/4 (45 degrees):

  sin(2 * π/4) = sin(π/2) = 1

  2sin(π/4) = 2 * (√2 / 2) = √2

  Since 1 ≠ √2, the equation is not an identity.

Example 14: Is (p + q)² - (p - q)² = 4pq an identity?

• Method: Simplification/Expansion:

  (p + q)² = p² + 2pq + q²

  (p - q)² = p² - 2pq + q²

  Therefore:

  (p + q)² - (p - q)² = (p² + 2pq + q²) - (p² - 2pq + q²) = p² + 2pq + q² - p² + 2pq - q² = 4pq

  This matches the right side. Therefore, the equation is an identity.

Example 15: Is √(x²) = x an identity?

• Method: Substitution. Consider x = -3:

  √((-3)²) = √(9) = 3

  Since 3 ≠ -3, the equation is not an identity. The correct identity is √(x²) = |x|.

Example 16: Is (1 + tan²(θ)) = sec²(θ) an identity?

• Method: This is a standard trigonometric identity derived from the Pythagorean identity. Therefore, the equation is an identity.

Example 17: Is cos(a + b) = cos(a) + cos(b) an identity?

• Method: Substitution. Let a = π/2 and b = 0:

  cos(π/2 + 0) = cos(π/2) = 0

  cos(π/2) + cos(0) = 0 + 1 = 1

  Since 0 ≠ 1, the equation is not an identity. The correct identity is cos(a + b) = cos(a)cos(b) - sin(a)sin(b).

Example 18: Is x³ - y³ = (x - y)(x² + xy + y²) an identity?

• Method: Simplification/Expansion. Expanding the right side:

  (x - y)(x² + xy + y²) = x³ + x²y + xy² - x²y - xy² - y³ = x³ - y³

  This matches the left side. Therefore, the equation is an identity.

Example 19: Is (a + b + c)² = a² + b² + c² an identity?

• Method: Substitution. Let a = 1, b = 1, c = 1:

  (1 + 1 + 1)² = 3² = 9

  1² + 1² + 1² = 1 + 1 + 1 = 3

  Since 9 ≠ 3, the equation is not an identity. The correct expansion is (a + b + c)² = a² + b² + c² + 2ab + 2ac + 2bc.

Example 20: Is sin(θ + π) = -sin(θ) an identity?

• Method: This is a trigonometric identity related to the periodicity and symmetry of the sine function. Therefore, the equation is an identity.

The Importance of Context and Restrictions

While many identities hold true for all possible values of the variables, some are subject to restrictions. For example, the logarithmic identity log(xy) = log(x) + log(y) is only valid for x > 0 and y > 0. Similarly, trigonometric identities involving tangent or secant functions are undefined at angles where cosine is zero. It is crucial to be aware of these restrictions when working with identities. Failing to account for them can lead to incorrect results and a misunderstanding of the underlying mathematical principles. Therefore, when determining if an equation is an identity, always consider the domain of the functions involved and any potential restrictions on the variables.

Conclusion

Understanding and identifying identities is a fundamental skill in algebra and trigonometry. The ability to recognize and manipulate identities allows for simplification, problem-solving, and a deeper appreciation of mathematical relationships. By employing techniques like simplification, substitution, expansion, and factoring, we can confidently determine whether an equation is an identity and leverage its properties to our advantage. Furthermore, recognizing the importance of context and restrictions ensures accurate and meaningful mathematical analysis. Mastering identities is an investment in mathematical fluency that pays dividends in more advanced studies.