📚 Proving Trigonometric Identities | 证明三角恒等式
Proving trigonometric identities is one of the most important skills in the Edexcel A Level Mathematics syllabus. It combines algebraic manipulation, knowledge of exact trigonometric values and a deep understanding of how different trigonometric functions relate to one another. Many students find these questions challenging at first, but with a clear strategy and fluency in the core identities, they become very manageable.
证明三角恒等式是 Edexcel A Level 数学大纲中最重要的技能之一。它结合了代数变形、特殊角的精确值以及不同三角函数之间关系的深入理解。许多学生一开始觉得这类题目很难，但只要掌握清晰的解题策略并熟练运用核心恒等式，这类题目就会变得非常容易应对。
1. What is a trigonometric identity? | 什么是三角恒等式？
erent from a trigonometric equation such as sin θ = 0.5, which has only certain solutions.
在 Edexcel A Level 数学中，三角恒等式是包含三角函数的命题，对于所有使两边有定义的变量值都成立。这与 sin θ = 0.5 这类只在某些角度成立的三角方程不同。
When a question says 'prove that ...', you are asked to manipulate one side of the identity, usually the more complicated side, until it is identical to the other side. Alternatively, you may transform both sides separately to the same expression, but you must show clear algebraic steps.
当题目要求 '证明……' 时，你需要对恒等式的一边（通常是较复杂的一边）进行变形，直到它与另一边完全相同。也可以将两边分别化为同一个表达式，但必须展示清晰的代数步骤。
does not prove an identity. You must use symbolic manipulation so that the equality is established for all valid values of θ.
证明与代入几个特殊值验证不同。即使两边在 θ = 30° 和 θ = 45° 时得到相同结果，也不能证明一个恒等式。你必须使用符号变形，使等式对所有有效的 θ 值都成立。
2. Essential identities to learn | 必须掌握的基本恒等式
Before attempting proof questions, you must be able to recall the core identities quickly. The table below summarises the reciprocal, quotient and Pythagorean identities that appear constantly in Edexcel papers.
在尝试证明题之前，你必须能够快速回忆起核心恒等式。下表总结了 Edexcel 试卷中经常出现的倒数、商数和毕达哥拉斯恒等式。
| Identity | Name |
|---|---|
| sin²θ + cos²θ = 1 | Pythagorean identity |
| 1 + tan²θ = sec²θ | Pythagorean identity |
| 1 + cot²θ = cosec²θ | Pythagorean identity |
| tan θ = sin θ / cos θ | Quotient identity |
| cosec θ = 1 / sin θ | Reciprocal identity |
| sec θ = 1 / cos θ | Reciprocal identity |
| cot θ = cos θ / sin θ = 1 / tan θ | Reciprocal / quotient identity |
You are expected to use these identities fluently. In particular, rewriting tan θ, cot θ, sec θ and cosec θ in terms of sin θ and cos θ is often the first step in a proof.
你需要熟练运用这些恒等式。特别是将 tan θ、cot θ、sec θ 和 cosec θ 用 sin θ 和 cos θ 表示，通常是证明题的第一步。
3. Pythagorean identities | 毕达哥拉斯恒等式
by cos²θ gives 1 + tan²θ = sec²θ. Dividing every term by sin²θ gives 1 + cot²θ = cosec²θ.
最重要的恒等式是 sin²θ + cos²θ = 1。各项同时除以 cos²θ 可得到 1 + tan²θ = sec²θ。各项同时除以 sin²θ 可得到 1 + cot²θ = cosec²θ。
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
= 1 − sin²θ and sin²θ = 1 − cos²θ allow you to replace a squared term with a single term.
对毕达哥拉斯恒等式进行移项通常是化简表达式的关键。例如，cos²θ = 1 − sin²θ 和 sin²θ = 1 − cos²θ 可以将一个平方项替换为一个单项。
This is especially useful when one side of an identity contains a denominator like 1 − sin²θ or 1 + tan²θ. Replacing those with cos²θ or sec²θ often leads to immediate cancellation.
当恒等式的一边含有 1 − sin²θ 或 1 + tan²θ 这样的分母时，这种替换尤其有用。将它们替换为 cos²θ 或 sec²θ 往往可以立即约分。
4. Addition formulae | 和差公式
The addition formulae are given in the Edexcel formula booklet, but you still need to recognise when to use them. They are especially useful when proving identities involving sums or differences of angles.
和差公式在 Edexcel 公式手册中给出，但你仍需要识别何时使用它们。它们在证明含有角度和或差的恒等式时尤为有用。
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
pposite sign between the two product terms. For sine, the sign stays the same as in A ± B.
使用 cos(A + B) 或 cos(A − B) 时要注意符号反转：余弦的两项乘积之间符号相反。对于正弦，符号与 A ± B 中的符号保持一致。
sin(A + B) may require expanding the sine first, then simplifying using sin²A + cos²A = 1.
许多考题会将和差公式与毕达哥拉斯恒等式结合。例如，证明含有 sin(A + B) 的恒等式时，可能需要先展开正弦，再利用 sin²A + cos²A = 1 进行化简。
5. Double angle identities | 二倍角恒等式
B. They are extremely common in proof questions and are also essential for integration and differentiation later in the course.
二倍角公式由和差公式令 A = B 得到。它们在证明题中极为常见，在后续课程的积分和微分中也是必不可少的。
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A
cos 2A = 2cos²A − 1
cos 2A = 1 − 2sin²A
tan 2A = (2 tan A) / (1 − tan²A)
There are three forms for cos 2A. Choosing the right form can make a proof much shorter: use 1 − 2sin²A when you want to introduce sine terms, and use 2cos²A − 1 when you want to introduce cosine terms.
cos 2A 有三种形式。选择正确的形式可以大大缩短证明过程：当你需要引入正弦项时，使用 1 − 2sin²A；当你需要引入余弦项时，使用 2cos²A − 1。
sin²A is often the fastest substitution. Likewise, 1 + cos 2A is often replaced by 2cos²A.
例如，如果恒等式中出现 1 − cos 2A，使用 1 − (1 − 2sin²A) = 2sin²A 通常是最快的替换方式。同样，1 + cos 2A 常常替换为 2cos²A。
6. General proving strategies | 一般证明策略
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图