Trigonometric identities and proofs Cambridge IGCSE Additional Mathematics revision
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In plain words
An identity is an equation that is true for every angle. A handful of them let you turn one trigonometric expression into another, and "prove that" questions are exercises in doing exactly that, one clear step at a time.
6 things to know
- tan x = sin x ÷ cos x, and cot x = cos x ÷ sin x.
- sin²x + cos²x = 1. It can be rearranged: 1 − sin²x = cos²x, and 1 − cos²x = sin²x.
- sec²x = 1 + tan²x, and cosec²x = 1 + cot²x. All three squared identities are on the formula list.
- To prove an identity, start with one side, usually the more complicated one, and change it step by step until it becomes the other side.
- Useful moves: write everything in terms of sin and cos; put fractions over a common denominator; look for sin²x + cos²x.
- Never move terms across the equals sign in a proof. That treats the result as already true.
Worked example
Prove that sin x tan x + cos x = sec x.
- Write tan x as sin x ÷ cos x: the left-hand side is sin²x ÷ cos x + cos x.
- Put it over one denominator: (sin²x + cos²x) ÷ cos x.
- The top is 1, so this is 1 ÷ cos x, which is sec x.
Tips and tricks
- Write "LHS =" at the start and keep one side only on the page. Finish with "= RHS".
- Every step must be visible. The answer is given, so the marks are all for the working.
It lands in your notebook with its questions as flashcards.
Trigonometric identities and proofs: 6 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is sin²x + cos²x equal to?
It is true for every angle.
What is 1 + tan²x equal to?
One of the identities on the formula list.
What is 1 + cot²x equal to?
One of the identities on the formula list.
What is 1 − sin²x equal to?
Rearranged from sin²x + cos²x = 1.
What is sec²x − 1 equal to?
Rearranged from sec²x = 1 + tan²x.
What is sin x ÷ cos x equal to?
The definition of tangent in terms of sine and cosine.
Quiz
6 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 9 marks. Write your answers on paper, then check them.
Trigonometric identities and proofs
Cambridge IGCSE Additional Mathematics 0606 · 9 marks · papermunch.org
Name ______________________________ Date ______________
Prove that (1 + tan²x) cos²x = 1.[3]
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LHS = sec²x × cos²x = (1 ÷ cos²x) × cos²x = 1 = RHS.
Prove that cosec x − sin x = cos x cot x.[3]
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LHS = 1/sin x − sin x = (1 − sin²x)/sin x = cos²x/sin x = cos x × (cos x/sin x) = cos x cot x = RHS.
Prove that tan x + cot x = sec x cosec x.[3]
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LHS = sin x/cos x + cos x/sin x = (sin²x + cos²x)/(sin x cos x) = 1/(sin x cos x) = sec x cosec x = RHS.
Answers: Trigonometric identities and proofs
- 1. LHS = sec²x × cos²x = (1 ÷ cos²x) × cos²x = 1 = RHS.
- 2. LHS = 1/sin x − sin x = (1 − sin²x)/sin x = cos²x/sin x = cos x × (cos x/sin x) = cos x cot x = RHS.
- 3. LHS = sin x/cos x + cos x/sin x = (sin²x + cos²x)/(sin x cos x) = 1/(sin x cos x) = sec x cosec x = RHS.