Prove and simplify trigonometric identities with restrictions
Use the two basic identities to rewrite expressions, justify each algebraic step and preserve excluded angles.
Pure Mathematics 1 (Paper 1) · AS & A Level Mathematics 9709, 2026–2027. State angle units and find every solution in the supplied interval; general forms of solution are not required in section 1.5. Written practice is self-checked.
Before you begin
- Simplify fractions and factorise a difference of squares.
- Understand why division by zero is undefined.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does sin² θ mean?
What you will learn
- Use tan θ = sin θ/cos θ and sin² θ + cos² θ = 1.
- Prove or simplify expressions while retaining their original denominator restrictions.
Two identities connect the three ratios
On the unit circle, the coordinates (cos θ, sin θ) satisfy x² + y² = 1. Therefore sin² θ + cos² θ = 1 for every real θ. Rearranging gives sin² θ = 1 − cos² θ or cos² θ = 1 − sin² θ. The notation sin² θ means (sin θ)², not sin(θ²) or sin(2θ).
Where cos θ ≠ 0, tan θ = sin θ/cos θ. Substitute this quotient to place expressions over a common denominator, then use the squared identity. For example tan θ cos θ = sin θ on the original tangent domain. The simplified sine is defined at more angles, but the original expression remains undefined where cosine is zero.
A proof works for every angle in the stated domain
To prove an identity, rewrite one side through valid equalities until it matches the other. Checking a few numerical angles can detect an error but cannot prove an identity for every permitted angle. State which denominators must be non-zero before cancelling. For (1 − cos² θ)/sin θ, replace the numerator by sin² θ and cancel one factor to get sin θ, retaining sin θ ≠ 0.
For sin θ/cos θ + cos θ/sin θ, use denominator sin θ cos θ to obtain (sin² θ + cos² θ)/(sin θ cos θ) = 1/(sin θ cos θ). Both sine and cosine must be non-zero. Cancellation removes a factor, not the need to respect the original domain. An identity is a statement valid throughout its domain; an equation asks which particular angles satisfy it.
Pause and explain
(1 − cos² θ)/sin θ simplifies to sin θ. Which restriction remains?
Worked example
Simplify (1 − cos² θ)/sin θ and state its restriction.
Show the worked solution
- The original denominator requires sin θ ≠ 0.
- Use 1 − cos² θ = sin² θ to rewrite the fraction as sin² θ/sin θ.
- Cancel one non-zero sine factor to obtain sin θ, retaining the original restriction.
Answer sin θ, for sin 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.
Simplify (1 − sin² θ)/cos θ and retain its domain restriction.
Give me a hint
Use the squared identity before cancelling a factor.
Compare my reasoning
- The denominator requires cos θ ≠ 0.
- Replace 1 − sin² θ by cos² θ.
- Then cos² θ/cos θ = cos θ on the original permitted domain.
cos θ, for cos θ ≠ 0.
Look for these in your work
- Use the correct rearrangement of the squared identity.
- Keep the original denominator restriction.
Prove sin θ/cos θ + cos θ/sin θ = 1/(sin θ cos θ) where both sides are defined.
Give me a hint
Combine the left fractions using the product denominator.
Compare my reasoning
- Require sin θ ≠ 0 and cos θ ≠ 0.
- The left side becomes (sin² θ + cos² θ)/(sin θ cos θ).
- Apply sin² θ + cos² θ = 1 to obtain the right side throughout that domain.
Identity proved for sin θ ≠ 0 and cos θ ≠ 0.
Look for these in your work
- Combine the fractions with their correct numerators.
- Justify the identity on the common domain.
Prove (1 − cos θ)/sin θ = sin θ/(1 + cos θ) where both original expressions are defined. Explain why θ = 0 cannot be restored by cancellation.
Give me a hint
Multiply the left numerator and denominator by 1 + cos θ, then factor or use the squared identity.
Compare my reasoning
- The common domain requires sin θ ≠ 0 and 1 + cos θ ≠ 0.
- The left side becomes (1 − cos² θ)/(sin θ(1 + cos θ)) = sin² θ/(sin θ(1 + cos θ)).
- Cancel non-zero sine to obtain the right side. At θ = 0, the original left fraction is 0/0, so it remains undefined although the right side is 0.
Identity proved on the common domain; θ = 0 is excluded by the original left denominator.
Look for these in your work
- Use a valid multiplier and the difference-of-squares identity.
- Explain why an algebraically simplified value does not remove an original exclusion.
Common mistakes
- sin² θ is the square of the sine output, not a doubled angle.
- Numerical spot checks are not a general proof.
- A cancelled denominator still supplies an original-domain restriction.
What can you explain now?
Simplify tan θ cos θ and state where the original expression is defined.
Compare with the explanation
sin θ, with cos θ ≠ 0.
Replace tangent by sine divided by cosine, cancel the non-zero cosine and retain the restriction.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Prove one identity by transforming just one side. Mark each division or cancellation and state the angle restrictions it requires.
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.