Trigonometric identity errors are often not caused by forgetting an identity. They come from unsafe algebra, hidden restrictions or reasoning that quietly assumes the result being proved.
1. Circular reasoning
If the target is A=B, beginning with “A=B” and manipulating until a true statement appears does not necessarily prove the original identity. The argument may have assumed what it needed to establish.
A safer proof starts from one side and transforms through known equalities until it reaches the other side.
2. Invalid cancellation across addition
In (sin x+cos x)/sin x, sin x cannot be cancelled from only one term of the numerator.
Instead split the fraction, where defined: 1+cot x. Cancellation requires a common factor of the whole numerator and denominator.
3. Hidden restrictions from denominators
An expression containing 1/cos x is undefined when cos x=0. A proof may simplify correctly elsewhere but still be stated too broadly if the original domain is ignored.
Write identities as valid where both sides are defined.
4. Dividing by an expression that might be zero
If a proof divides by sin x, it assumes sin x≠0. That may be acceptable when the original expression already has sin x in a denominator. It is unsafe if the original identity also includes values where sin x=0.
Factorisation can sometimes avoid introducing an unnecessary restriction.
5. Misusing Pythagorean identities
From sin²x+cos²x=1, valid rearrangements include 1−sin²x=cos²x and 1−cos²x=sin²x.
It does not imply sin x+cos x=1. Squared terms cannot be unsquared term-by-term in that way.
6. Confusing an identity with an equation
sin²x+cos²x=1 is an identity. sin x=1/2 is an equation whose truth depends on x. Proof questions and solution questions require different goals.
7. Replacing a function with the wrong reciprocal
sec x=1/cos x, cosec x=1/sin x and cot x=1/tan x=cos x/sin x where defined. Swapping reciprocal pairs changes the expression fundamentally.
8. Losing brackets during substitution
If tan x=sin x/cos x, then tan²x=(sin x/cos x)²=sin²x/cos²x. Squaring only the numerator or denominator is incorrect.
9. Treating numerical checks as proof
Testing x=30° and x=45° can expose an error, but agreement at a few values does not prove an identity for every allowed x. Numerical checks support verification; the symbolic chain provides the proof.
10. A repair routine
- Find the first line that is not guaranteed by the previous line.
- Check whether a denominator or cancellation introduced a restriction.
- Check whether an algebraic factor was mistaken for a term.
- Return to a known identity and rebuild from one side.
- Use one numerical value only as a secondary check.
11. Practice diagnosis
- A student writes (sin x+1)/sin x=1+1. Identify the error.
- A student divides sin x cos x=sin x by sin x without comment. What condition has been introduced?
- A student writes sin²x+cos²x=1, therefore sin x+cos x=1. Explain the error.
- A student proves an identity by checking x=0°,30°,60°. Is this sufficient?
- State the domain restriction of tan x=sin x/cos x.
12. Answers
- sin x is not a factor of the entire numerator. The correct split is 1+cosec x, where defined.
- sin x≠0.
- The identity involves squares; taking square roots term-by-term is invalid.
- No. A finite set of checks cannot prove the identity for every allowed x.
- cos x≠0.
Continue with How to Prove Trigonometric Identities Without Guessing, Factorisation, Common Denominators and Substitution, or return to the Additional Mathematics Hub.