Proving Trigonometric Identities: Strategies and Methods | 三角恒等式证明的思路与方法

📚 Proving Trigonometric Identities: Strategies and Methods | 三角恒等式证明的思路与方法

Trigonometric identities are equations that hold true for all values of the variable within their domain. Proving them requires a combination of algebraic manipulation, knowledge of fundamental identities, and strategic thinking. This article explores the key approaches and techniques used to verify trigonometric identities efficiently and rigorously.

三角恒等式是在其定义域内对变量所有取值均成立的等式。证明三角恒等式需要代数变形、对基本恒等式的熟练掌握以及策略性思维。本文将系统探讨验证三角恒等式的主要思路与技巧。

1. Start from the More Complex Side | 从较复杂的一边开始

The most common strategy is to begin with the side of the identity that looks more complicated, then simplify it step by step until it matches the other side. This is usually more straightforward because complex expressions offer more opportunities for substitution and simplification.

最常用的策略是从恒等式中看起来较复杂的一边入手，再逐步化简，直到与另一边相等。这样做通常更直接，因为复杂的表达式为代换和化简提供了更多机会。

Choose the side with more terms, fractions, or mixed functions.

选择项数更多、含分式或混合函数更多的一边。
Keep the simpler side fixed as your target.

保持较简单的一边作为目标不变。

Example: Prove tan x + cot x = sec x csc x.

Start with the left side: tan x + cot x = sin x/cos x + cos x/sin x = (sin² x + cos² x)/(sin x cos x) = 1/(sin x cos x) = sec x csc x.

从左式出发：tan x + cot x = sin x/cos x + cos x/sin x = (sin² x + cos² x)/(sin x cos x) = 1/(sin x cos x) = sec x csc x。

2. Master the Fundamental Identities | 熟练掌握基本恒等式

Every proof ultimately relies on a small set of core identities. You must know them so well that you can recognise their presence in various forms without hesitation.

任何证明最终都依赖少数核心恒等式。你必须对这些恒等式了然于胸，以便在各种形式中毫不犹豫地识别它们。

Reciprocal 倒数	csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x
Quotient 商数	tan x = sin x/cos x, cot x = cos x/sin x
Pythagorean 平方	sin² x + cos² x = 1; tan² x + 1 = sec² x; 1 + cot² x = csc² x
Even/Odd 奇偶	sin(-x) = -sin x, cos(-x) = cos x, tan(-x) = -tan x

When you see a squared trigonometric function, think of the Pythagorean identities. When you see a fraction like sin/cos, think of the tangent.

看到平方三角函数，立即联想到平方恒等式；看到 sin/cos 这样的分式，联想到正切。

3. Convert Everything to Sine and Cosine | 全部化为正弦和余弦

A highly effective strategy is to rewrite all trigonometric functions in terms of sin x and cos x. This eliminates the need to remember separate identities for tan, cot, sec, and csc, and reduces the expression to algebraic fractions.

一个非常有效的策略是将所有三角函数都写成 sin x 和 cos x 的形式。这样就无需单独记忆 tan、cot、sec、csc 的恒等式，并将表达式化为代数分式。

Replace tan x = sin x/cos x, sec x = 1/cos x, etc.

用 tan x = sin x/cos x、sec x = 1/cos x 等替换。
Then combine fractions and simplify using sin² x + cos² x = 1.

然后合并分式，并利用 sin² x + cos² x = 1 化简。

Example: Prove (1 + cot x) / (1 + tan x) = cot x.

Rewrite as (1 + cos/sin) / (1 + sin/cos) = ((sin+cos)/sin) / ((cos+sin)/cos) = (sin+cos)/sin × cos/(sin+cos) = cos/sin = cot x.

改写为 (1 + cos/sin) / (1 + sin/cos) = ((sin+cos)/sin) / ((cos+sin)/cos) = (sin+cos)/sin × cos/(sin+cos) = cos/sin = cot x。

4. Multiply by Conjugates | 利用共轭式相乘

When an expression contains a sum or difference of square roots, or a binomial involving 1 ± sin x or 1 ± cos x, multiplying numerator and denominator by the conjugate can create a difference of squares that simplifies to a Pythagorean identity.

当表达式包含平方根的和差，或含有 1 ± sin x、1 ± cos x 的二项式时，在分子分母同乘共轭式可构造平方差，从而转化为平方恒等式进行化简。

Example: Prove 1/(1 + sin x) = sec x – tan x.

Multiply numerator and denominator by 1 – sin x: (1 – sin x)/[(1 + sin x)(1 – sin x)] = (1 – sin x)/(1 – sin² x) = (1 – sin x)/cos² x. This does not simplify directly, but if we instead start from the right side: sec x – tan x = 1/cos x – sin/cos x = (1 – sin x)/cos x. Then multiply numerator and denominator by (1 + sin x) to get (1 – sin² x)/[cos x(1+sin x)] = cos² x/[cos x(1+sin x)] = cos x/(1+sin x). Wait, this gives cos/(1+sin), which is not equal to 1/(1+sin) unless cos=1. Let’s correct: Actually the identity is 1/(1 + sin x) = sec x – tan x? Check x=30°, LHS=1/(1.5)=0.666, RHS=1.154-0.577=0.577. Not true. The correct identity is sec x – tan x = tan(π/4 – x/2) or = (1 – sin x)/cos x. To show 1/(1+sin x) = sec x – tan x is false. Let’s choose a correct example: Prove (sec x + tan x) = 1/(sec x – tan x). Or prove 1/(1+cos x) = csc² x – csc x cot x? Better: Prove (1 – cos x)/sin x = sin x/(1 + cos x). Multiply numerator and denominator of left by (1+cos x)? Actually this is a classic. Let’s use that.

注意：恒等式 1/(1+sin x)=sec x – tan x 并不成立。我们改用经典例子：证明 (1 – cos x)/sin x = sin x/(1 + cos x)。这可以通过在左边分子分母同乘 (1+cos x) 来证明：(1 – cos x)(1+cos x)/[sin x(1+cos x)] = (1 – cos² x)/[sin x(1+cos x)] = sin² x/[sin x(1+cos x)] = sin x/(1+cos x)。

5. Combine and Split Fractions | 分式的合并与拆分

Many identities involve sums of fractions. Combining fractions by finding a common denominator often yields a numerator that simplifies to a basic identity. Conversely, splitting a single fraction into two parts can match the target expression.

许多恒等式涉及分式之和。通过通分合并分式，往往能得到可化简为基本恒等式的分子。反之，将单项分式拆成两项也能与目标表达式匹配。

For sums: a/b + c/d = (ad+bc)/bd.

求和：a/b + c/d = (ad+bc)/bd。
For products: tan x sin x = sin² x/cos x, etc.

求积：tan x sin x = sin² x/cos x 等。

Example: Prove 1/(1 – sin x) + 1/(1 + sin x) = 2 sec² x.

Combine: [(1+sin x)+(1-sin x)]/[(1-sin x)(1+sin x)] = 2/(1 – sin² x) = 2/cos² x = 2 sec² x.

合并：[(1+sin x)+(1-sin x)]/[(1-sin x)(1+sin x)] = 2/(1 – sin² x) = 2/cos² x = 2 sec² x。

6. Use Double-Angle and Half-Angle Identities | 运用二倍角与半角恒等式

When the variable appears as 2x or x/2, double-angle and half-angle identities are essential. They also help to reduce powers: sin² x = (1 – cos 2x)/2, cos² x = (1 + cos 2x)/2.

当变量以 2x 或 x/2 出现时，二倍角与半角恒等式至关重要。它们还能用于降幂：sin² x = (1 – cos 2x)/2，cos² x = (1 + cos 2x)/2。

sin 2x = 2 sin x cos x	cos 2x = cos² x – sin² x = 2 cos² x – 1 = 1 – 2 sin² x
tan 2x = 2 tan x / (1 – tan² x)	sin² x = (1 – cos 2x)/2, cos² x = (1 + cos 2x)/2

Example: Prove sin 2x / (1 + cos 2x) = tan x.

Using sin 2x = 2 sin x cos x and 1 + cos 2x = 2 cos² x, we get 2 sin x cos x / (2 cos² x) = sin x/cos x = tan x.

利用 sin 2x = 2 sin x cos x 和 1 + cos 2x = 2 cos² x，得 2 sin x cos x / (2 cos² x) = sin x/cos x = tan x。

7. Apply Sum-to-Product and Product-to-Sum Formulas | 应用和差化积与积化和差公式

When an identity contains sums or products of sine and cosine with different angles, converting them using these formulas can reveal common factors or simplify the expression.

当恒等式含有不同角的正弦、余弦之和或乘积时，用这些公式进行转化可以揭示公因式或简化表达式。

Product-to-sum: 2 sin A sin B = cos(A-B) – cos(A+B)

积化和差：2 sin A sin B = cos(A-B) – cos(A+B)
Sum-to-product: sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)

和差化积：sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)

Example: Prove (sin 3x + sin x) / (cos 3x + cos x) = tan 2x.

Sum-to-product: numerator = 2 sin 2x cos x; denominator = 2 cos 2x cos x. The ratio = sin 2x/cos 2x = tan 2x.

和差化积：分子 = 2 sin 2x cos x；分母 = 2 cos 2x cos x。比值 = sin 2x/cos 2x = tan 2x。

8. Prove by Transforming Both Sides | 两边同时变形的证明方法

When neither side is obviously simpler, you can simplify both sides independently until they reach the same intermediate expression. This is a legitimate proof as long as each step is reversible.

当两边都没有明显更简单时，可以分别化简两边，直到它们得到同一个中间表达式。只要每一步都可逆，这就是合法的证明。

Write the two sides as separate columns and work them downwards toward a common expression. Avoid treating the identity as an equation to solve; do not move terms across the equals sign without justification.

把两边分别写在两列中，向下推导至共同表达式。避免把恒等式当作方程来解；不能无理由地将项移到等号另一边。

Example: Prove (1 – tan x)/(1 + tan x) = (1 – sin 2x)/cos 2x.

Left: (cos – sin)/(cos + sin). Right: multiply numerator and denominator of left? Actually right: (1 – sin 2x)/cos 2x = (cos – sin)²/(cos² – sin²) = (cos – sin)/(cos + sin). Both sides equal.

左边：(cos – sin)/(cos + sin)。右边：(1 – sin 2x)/cos 2x = (cos – sin)²/(cos² – sin²) = (cos – sin)/(cos + sin)。两边相等。

9. Check the Domain and Avoid Invalid Steps | 注意定义域并避免无效步骤

Trigonometric identities are not valid where any function in the identity is undefined. For example, tan x is undefined when cos x = 0, so the identity sec x tan x = sin x sec² x holds only where all terms are defined.

三角恒等式在式中任一函数无定义处不成立。例如，tan x 在 cos x = 0 时无定义，因此恒等式 sec x tan x = sin x sec² x 仅在所有项都有定义时成立。

Avoid dividing by zero; if you divide by a trigonometric expression, note where it is nonzero.

避免除以零；若除以某个三角表达式，需注明其非零条件。
Do not square both sides unless you check for extraneous signs.

不要急于两边平方，除非检查增根导致的符号问题。

When proving an identity, state that the variable is restricted to values for which all involved expressions are defined.

证明恒等式时，应声明变量取使所有相关表达式都有定义的值。

10. Practice with a Systematic Algorithm | 用系统化流程进行练习

For difficult identities, follow this step-by-step algorithm to keep your thinking organised.

对较难的恒等式，按下列系统化流程保持思路清晰。

Identify the target: choose the side to simplify, usually the more complex one.

确定目标：选择要化简的一边，通常是较复杂的一边。
Convert all functions to sine and cosine if other functions do not simplify immediately.

若其他函数无法立即化简，则将一切函数化为正弦与余弦。
Use Pythagorean identities to replace 1, or to change squares.

利用平方恒等式替换 1，或者变换平方项。
Combine fractions, factor, cancel common factors.

合并分式、因式分解、约去公因式。
If stuck, try rewriting using double-angle or sum-to-product formulas.

若卡住，尝试用二倍角或和差化积公式重写。

Practice with a variety of problems. Once you recognise patterns, choosing the right strategy becomes automatic.

通过多样化的题目进行练习。一旦你识别出模式，选择正确策略就变得自然而然。

By combining these strategies, you can approach any trigonometric identity proof with confidence. The key is flexibility: try one method, and if it leads nowhere, switch to another. Remember that algebra is the foundation; the more fluently you can manipulate fractions and factor expressions, the easier proofs become.

综合运用这些策略，你可以自信地应对任何三角恒等式证明。关键在于灵活：一种方法行不通就换另一种。牢记代数是基础；你越能熟练地进行分式运算和因式分解，证明就越容易。

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Proving Trigonometric Identities: Strategies and Methods | 三角恒等式证明的思路与方法