Prove trigonometric relationships using identities
Build a proof from one side, choose an identity that simplifies it and state any restrictions needed for the result to hold.
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What identity can replace sin²θ + cos²θ?
What you will learn
- Prove a trigonometric relationship by transforming one side into the other
- Choose an identity that reduces the number of different functions
- State restrictions that keep every denominator nonzero
Start with one side and make a reasoned change
Choose the side with more terms or more function types. Rewrite sec, cosec, tan or cot as ratios when useful, or replace a squared expression with one of the supplied identities.
Show each algebraic step and keep the equality true. Do not assume the target statement while simplifying, and do not transform both sides into an expression whose equality has not been established.
Every cancellation has a domain condition
For example, sin θ tan θ + cos θ = sec θ can be proved by replacing tan θ with sin θ/cos θ, then using sin²θ + cos²θ = 1. This requires cos θ ≠ 0, which is also the domain of tan and sec.
If a proof divides by sin θ, state sin θ ≠ 0. If the problem gives an interval, check that the excluded angles are not included or handle them separately where both sides are defined.
Pause and explain
Why must cos θ ≠ 0 be stated when using tan θ = sin θ/cos θ?
Worked example
Prove sin θ tan θ + cos θ = sec θ for cos θ ≠ 0.
Show the worked solution
- Start with the left-hand side and replace tan θ by sin θ/cos θ.
- Combine over a common denominator: sin²θ/cos θ + cos²θ/cos θ = (sin²θ + cos²θ)/cos θ.
- Use sin²θ + cos²θ = 1, giving 1/cos θ = sec θ, as required.
Answer The left-hand side simplifies to sec θ; the proof is valid where cos θ ≠ 0.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Prove (1 + tan²θ)cos²θ = 1 for cos θ ≠ 0.
Give me a hint
Replace 1 + tan²θ with sec²θ, then use sec θ = 1/cos θ.
Compare my reasoning
- Start with the left-hand side.
- Use 1 + tan²θ = sec²θ to obtain sec²θ cos²θ.
- Since sec θ cos θ = 1 where cosine is nonzero, the expression equals 1.
(1 + tan²θ)cos²θ = sec²θ cos²θ = 1, as required.
Look for these in your work
- Identity selected correctly
- Domain condition respected
- Target value reached from one side
Prove (cosec²θ − 1)sin θ = cot θ cos θ for sin θ ≠ 0.
Give me a hint
Replace cosecant squared minus one with cotangent squared, then express cotangent as cos/sin.
Compare my reasoning
- The left-hand side is cot²θ sin θ by cosec²θ − 1 = cot²θ.
- Write cot θ = cos θ/sin θ, so cot²θ sin θ = (cos²θ/sin²θ)sin θ = cos²θ/sin θ.
- The right-hand side is cot θ cos θ = (cos θ/sin θ)cos θ = cos²θ/sin θ; the sides agree.
Both sides equal cos²θ/sin θ, so the identity is proved for sin θ ≠ 0.
Look for these in your work
- Cosecant identity applied
- Both sides reduced with valid denominators
- Domain condition stated
A student writes (sec²θ − 1)/tan θ = tan θ and cancels tan θ without a condition. Repair the proof by stating its full domain.
Give me a hint
Check the definitions of sec and tan, then check that the cancelled factor is nonzero.
Compare my reasoning
- Use sec²θ − 1 = tan²θ, so the expression is tan²θ/tan θ.
- Cancellation requires tan θ ≠ 0; sec and tan also require cos θ ≠ 0.
- Together these conditions require sin θ ≠ 0 and cos θ ≠ 0.
It equals tan θ only where sin θ ≠ 0 and cos θ ≠ 0; at other angles the original expression is undefined or the cancellation is invalid.
Look for these in your work
- Identity used
- Cancelled factor restriction retained
- Original function domain included
Common mistakes
- Starting by assuming the equality to be proved
- Changing both sides without recording equivalent steps
- Cancelling a factor that may equal zero
- Omitting the domain restriction on secant, tangent, cosecant or cotangent
What can you explain now?
Which side is usually safest to transform in an identity proof?
Compare with the explanation
One side, chosen for simpler reduction
Transforming one side lets each equality be justified until it matches the other side.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How can you prove a trigonometric identity from one side while keeping track of the values at which each step is valid?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.