📚 Identities and Equations | 恒等式与方程
In algebra, the concepts of identities and equations form the backbone of mathematical reasoning. An identity is a statement that is true for all permissible values of the variables involved, while an equation is a statement that is only true for certain specific values, known as solutions. Mastering this distinction is crucial for solving problems ranging from polynomial manipulations to trigonometric proofs in the IB Mathematics curriculum. This article explores the definitions, notation, types, and common techniques associated with identities and equations.
在代数中，恒等式与方程的概念构成了数学推理的支柱。恒等式是对所涉及变量的所有允许值都成立的陈述，而方程是仅对某些特定值（即解）才成立的陈述。掌握这种区别对于解决 IB 数学课程中从多项式运算到三角证明的各类问题至关重要。本文探讨与恒等式和方程相关的定义、符号、类型和常用技巧。
1. Defining Identities and Equations | 定义恒等式与方程
ty, because evaluating both sides for any real number x always produces the same value. In IB Mathematics we often use the identity symbol ≡ phasise this fact, but a standard equals sign is also common in textbooks and exam papers.
恒等式是一种等式，对于其变量的每一个允许值都成立。例如，展开式 (x + 2)² = x² + 4x + 4 实际上是一个恒等式，因为对于任意实数 x，计算两边始终相等。在 IB 数学中，我们常用恒等符号 ≡ 来强调这一点，但在教材和试卷中也经常看到普通的等号。
ds only when x = 5. Solving an equation means finding these unknown values, and the solutions can be checked by substitution into the original statement.
另一方面，方程是仅对变量的特定值才成立的陈述。例如，2x + 5 = 15 是一个方程，因为只有当 x = 5 时它才成立。求解方程就是为了找出这些未知值，并且可以通过代入原式来检验解。
| Feature | Identity | Equation |
| Truth | For all variable values | For specific solutions |
| Symbol | ≡ (often =) | = |
| Example | (a + b)² ≡ a² + 2ab + b² | 2x + 3 = 7 |
| Number of solutions | Infinitely many (all in domain) | Finite, none, or infinite in special cases |
The table above summarises the key differences: an identity is universally true, while an equation requires solving to find the restricted set of values that satisfy it.
上表总结了关键区别：恒等式普遍成立，而方程需要求解以找到满足它的有限值集。
2. The Identity Symbol (≡) and Its Usage | 恒等符号及其使用
The symbol ≡, read as "is identically equal to", is used to express that two expressions are equivalent for all values of the variables within their common domain. In IB exam papers you may be asked to determine whether a given statement is an identity or an equation, and using ≡ correctly signals to the examiner that you understand the difference.
符号 ≡ 读作 "恒等于"，用于表示两个表达式在它们的公共定义域内对变量的所有取值都等价。在 IB 试卷中，你可能会被要求判断一个给定的陈述是恒等式还是方程，正确使用 ≡ 能向考官表明你理解了两者的区别。
A classic example is the Pythagorean identity sin²θ + cos²θ ≡ e this with tanθ = 1, which is an equation whose solutions are θ = π/4 + kπ for integer k.
一个经典的例子是毕达哥拉斯恒等式 sin²θ + cos²θ ≡ 1，它对每一个实数角 θ 都成立。与之对比，tanθ = 1 是一个方程，其解为 θ = π/4 + kπ，其中 k 为整数。
Be careful: the same symbol ≡ is also used for congruence in number theory, for example 17 ≡ 2 (mod 5). You must judge from the context whether ≡ means "identically equal" or "congruent modulo n".
要注意：同一个符号 ≡ 在数论中也用于表示同余，例如 17 ≡ 2 (mod 5)。你必须根据上下文判断 ≡ 表示 "恒等" 还是 "模 n 同余"。
3. Testing Whether a Statement Is an Identity | 判断一个陈述是否为恒等式
There are three standard techniques. First, expand and simplify both sides algebraically; if they reduce to exactly the same expression, the statement is an identity. Second, substitute a convenient value: if a single value fails, the statement is certainly not an identity. Third, compare coefficients of like terms, which is the most rigorous method for polynomial statements.
有三种标准方法。第一，用代数方法展开并化简两边；如果它们化为完全相同的表达式，则该陈述是恒等式。第二，代入一个方便的值：只要有一个值不成立，该陈述就一定不是恒等式。第三，比较同类项的系数，这是处理多项式陈述最严谨的方法。
x² = x is true for x = 0 and x = 1, but it is an equation, not an identity, because it fails for x = 2.
一个常见的陷阱是：测试两三个值，发现都成立，就断定该陈述是恒等式。这并不是证明。例如 x² = x 对 x = 0 和 x = 1 成立，但它是方程而非恒等式，因为当 x = 2 时它不成立。
4. Key Algebraic Identities | 重要代数恒等式
Certain identities appear so frequently that they should be memorised. The difference of two squares, the perfect square expansions, the difference and sum of cubes, and the expansion of (a + b)³ all belong to this core set.
有些恒等式出现得极其频繁，应当牢记。平方差、完全平方展开式、立方和与立方差，以及 (a + b)³ 的展开式都属于这一核心集合。
(a + b)² ≡ a² + 2ab + b²
(a − b)² ≡ a² − 2ab + b²
a² − b² ≡ (a + b)(a − b)
a³ + b³ ≡ (a + b)(a² − ab + b²)
a³ − b³ ≡ (a − b)(a² + ab + b²)
(a + b)³ ≡ a³ + 3a²b + 3ab² + b³
These identities allow you to factorise, expand and simplify expressions quickly, and they are the building blocks for completing the square, partial fractions and trigonometric manipulation later in the course.
这些恒等式让你能够快速因式分解、展开和化简表达式，它们也是后续配方法、部分分式与三角变形的基础构件。
5. Conditional Equations and Solution Sets | 条件方程与解集
s no real solutions, but it has two complex solutions, x = 2i and x = −2i. Always state the number system you are working in.
条件方程只对一个受限的取值集合成立，这个集合称为解集。方程的定义域极为重要：x² = −4 没有实数解，但有两个复数解 x = 2i 和 x = −2i。始终要说明你在哪个数系中工作。
h is satisfied by every real x. Recognising these degenerate cases saves time in the exam.
有些方程是矛盾方程，例如 x + 1 = x + 3，它完全无解。另一些则是伪装成方程的恒等式，例如 2(x + 3) = 2x + 6，它对所有实数 x 都成立。识别这些退化情形能在考试中节省时间。
When you square both sides of an equation, or multiply by an expression containing the unknown, you may introduce extraneous solutions. Always substitute your final answers back into the original equation to reject them.
当你在方程两边平方，或乘以含有未知数的表达式时，可能会引入增根。一定要把最终答案代回原方程检验，以剔除这些增根。
6. Solving Linear and Quadratic Equations | 求解一次与二次方程
If a = 0 and b ≠ 0 there is no solution; if a = 0 and b = 0 every value of x works, so the "equation" is actually an identity.
一次方程 ax + b = 0（a ≠ 0）有唯一解 x = −b/a。若 a = 0 且 b ≠ 0，则无解；若 a = 0 且 b = 0，则任意 x 都成立，此时这个 "方程" 其实是恒等式。
tells you the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives two complex conjugate roots.
二次方程 ax² + bx + c = 0 可以用因式分解、配方法或求根公式求解。判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 有两个不同实根，Δ = 0 有一个重根，Δ < 0 有两个共轭复根。
x = (−b ± √b² − 4ac) / 2a
α + β = −b/a αβ = c/a
Vieta's formulas, shown above, connect the roots α and β to the coefficients without solving the equation, and they are frequently examined in IB papers.
上面给出的韦达定理把根 α 与 β 直接与系数联系起来，无需解方程即可使用，在 IB 试卷中经常考查。
7. Identities in Quadratics: Comparing Coefficients | 二次式恒等：比较系数
When two polynomials are identically equal, the coefficients of corresponding powers of the variable must be equal. This principle is the engine behind many exam questions of the form "find the constants A and B such that ...".
当两个多项式恒等时，变量同次幂的系数必须相等。这一原理是众多 "求常数 A 和 B 使得……" 型考题的核心引擎。
Consider the identity A(x − 1) + B(x + 1) ≡ B)x + (B − A). Comparing coefficients yields A + B = 3 and B − A = 1, so B = 2 and A = 1.
考虑恒等式 A(x − 1) + B(x + 1) ≡ 3x + 1。展开左边得 (A + B)x + (B − A)。比较系数得到 A + B = 3 与 B − A = 1，因此 B = 2，A = 1。
A = 1. Both methods are acceptable; substitution is faster, comparison of coefficients is more systematic.
另一种方法是代入法：由于恒等式对所有 x 成立，取 x = 1 得 2B = 4，故 B = 2；取 x = −1 得 −2A = −2，故 A = 1。两种方法都被接受；代入法更快，比较系数法更有条理。
8. Partial Fractions and Identities | 部分分式与恒等式
Partial fractions are a direct application of polynomial identities. To decompose 1 / ((x − 1)(x + 2)) into the form A / (x − 1) + B / (x + 2), you set up an identity and then find A and B.
部分分式是多项式恒等式的直接应用。要把 1 / ((x − 1)(x + 2)) 分解为 A / (x − 1) + B / (x + 2) 的形式，你需要建立一个恒等式，然后求出 A 和 B。
1 / ((x − 1)(x + 2)) ≡ A / (x − 1) + B / (x + 2)
Multiplying through by (x − 1)(x + 2) gives the identity 1 ≡ + 2) + B(x − 1). Substituting x = 1 gives A = 1/3, and substituting x = −2 gives B = −1/3.
两边乘以 (x − 1)(x + 2) 得到恒等式 1 ≡ A(x + 2) + B(x − 1)。代入 x = 1 得 A = 1/3，代入 x = −2 得 B = −1/3。
This "cover-up" technique works because the identity must hold for every value in the domain, so any convenient value may be used. For repeated factors or improper fractions, you must extend the method by also comparing coefficients.
这种 "遮盖法" 之所以有效，是因为恒等式必须在定义域内对每个值都成立，因此可以选用任何方便的值。对于重因子或假分式，你需要通过比较系数来扩展这一方法。
9. Trigonometric Identities | 三角恒等式
Trigonometric identities are equalities that hold for every angle in the common domain. The three Pythagorean identities are the most important, followed by the compound-angle and double-angle formulas.
三角恒等式是对公共定义域内每个角都成立的等式。三个毕达哥拉斯恒等式最为重要，其次是和角公式与倍角公式。
sin²θ + cos²θ ≡ 1
1 + tan²θ ≡ sec²θ 1 + cot²θ ≡ cosec²θ
sin(A ± B) ≡ sinA cosB ± cosA sinB
cos(A ± B) ≡ cosA cosB ∓ sinA sinB
sin2θ ≡ 2 sinθ cosθ cos2θ ≡ cos²θ − sin²θ
To prove a trigonometric identity, start with the more complicated side and simplify it towards the other side. Work on one side only, and never move terms across the ≡ sign, because that presupposes the result you are trying to prove.
证明三角恒等式时，应从较复杂的一边开始，把它化简到另一边。只能处理单边，绝不要把项移到 ≡ 号的另一侧，因为那等于预设了你要证明的结论。
Other useful tools include expressing everything in terms of sin and cos, using a common denominator, and applying the difference of two squares to expressions such as cos²θ − sin²θ.
其他有用的工具包括：把所有式子都化为 sin 与 cos、使用公分母，以及对 cos²θ − sin²θ 这类表达式使用平方差。
10. Trigonometric Equations | 三角方程
In contrast to identities, trigonometric equations hold only for particular angles. Because trigonometric functions are periodic, such equations usually have infinitely many solutions, and an interval is normally specified.
与恒等式相反，三角方程只对特定的角成立。由于三角函数是周期函数，这类方程通常有无穷多个解，因此题目一般会指定一个区间。
θ < 2π. The general solution is θ = π/6 + 2kπ or θ = 5π/6 + 2kπ, where k is an integer.
例如，sinθ = 1/2 在区间 0 ≤ θ < 2π 内的主解为 θ = π/6 与 θ = 5π/6。通解为 θ = π/6 + 2kπ 或 θ = 5π/6 + 2kπ，其中 k 为整数。
olution for the whole argument, then divide by the coefficient of θ and list every solution inside the required interval.
对于 sin(nθ) = c 或 cos(2θ) = c 形式的方程，先对整个自变量求出通解，再除以 θ 的系数，并列出指定区间内的全部解。
sin²θ − sinθ − 1 = 0, which should be treated as quadratics in sinθ and then solved for θ.
始终检查区间用的是角度还是弧度，并注意像 2sin²θ − sinθ − 1 = 0 这样的二次三角方程，应把它看作关于 sinθ 的二次方程求解，再进一步求 θ。
11. Proving Identities vs Solving Equations | 证明恒等式与解方程的区别
The strategies for the two tasks are completely different, and confusing them is a major source of lost marks. Proving an identity is a demonstration that two expressions are always the same; solving an equation is a search for the values that make it true.
这两类任务的策略完全不同，混淆它们是失分的主要原因之一。证明恒等式是证明两个表达式始终相同；解方程是寻找使等式成立的那些值。
| Aspect | Proving an Identity | Solving an Equation |
| Goal | style="border:1px solid #ccc;padding
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