An invalid cancellation is a step where something is removed from the numerator and denominator of a fraction that is not a common factor. It is one of the most common ways to lose marks in trig algebra, because the wrong line looks tidy. This lesson gives you a rule, a quick test and a repair.
This lesson closes the sequence in trigonometric identities and follows stating excluded values.
What is the rule for cancelling?
You may cancel only a factor of the whole numerator and the whole denominator. A factor is something multiplied. A term is something added or subtracted. So:
- sin x cos x / cos x = sin x is fine, because cos x multiplies everything in the numerator.
- (sin x + cos x) / cos x is not sin x + 1. The cos x in the numerator is a term inside a sum.
Cancelling a factor also assumes it is not zero, which links to excluded values.
How do you check a suspicious step?
- Choose a test angle such as 30° or 45°, where sin and cos are easy.
- Evaluate the line before the step and the line after it.
- Compare. If the numbers are different, the step is invalid.
- Choose a second angle if they agree, since a single value can match by accident.
- Repair the step by factorising, splitting the fraction or using an identity.
Worked example
A student simplifies (sin²x + cos²x) / (sin x + cos x) like this:
Line 1: (sin²x + cos²x) / (sin x + cos x)
Line 2: sin x + cos x (cancelling “the squares”)
Diagnose with x = 30°. sin 30° = 0.5 and cos 30° ≈ 0.866, so sin x + cos x ≈ 1.366.
Line 1: the numerator is sin²x + cos²x = 1, so the value is 1 / 1.366 ≈ 0.732.
Line 2: 1.366.
They are different, so Line 2 is invalid. The squares are not common factors of the numerator and denominator: the numerator is a sum of squares, the bottom is a sum.
Repair: use the identity first. sin²x + cos²x = 1, so the expression is 1 / (sin x + cos x), which has no further simplification. Check: 1 / 1.366 ≈ 0.732.
Note that x = 90° would not have caught the error here: sin 90° + cos 90° = 1, and both lines give 1. That is why you choose a test angle that is not special.
The mistake to watch for
Mistaken working: (sin x + cos x) / cos x = sin x + 1
The student cancelled cos x from the numerator and denominator.
Test at x = 45°: the left side is (0.707 + 0.707) / 0.707 = 2, and sin 45° + 1 = 1.707.
They are different. The correction is to split the fraction: sin x / cos x + cos x / cos x = tan x + 1. At 45° this gives 1 + 1 = 2.
When a cancellation is allowed, the factor is shown, like this: (sin x cos x + cos²x) / cos x = cos x (sin x + cos x) / cos x = sin x + cos x, for cos x ≠ 0.
Check yourself
1. Is (sin x + 2) / 2 = sin x + 1? Test with x = 90° and correct it if necessary.
Show answer
At x = 90° the left side is (1 + 2) / 2 = 1.5, while sin x + 1 = 2. Not equal, so the step is invalid. The correct form is (sin x)/2 + 1, which gives 0.5 + 1 = 1.5.
2. Simplify (sin x cos x − cos²x) / cos x.
Show answer
Factorise the numerator: cos x (sin x − cos x) / cos x = sin x − cos x, valid for cos x ≠ 0.
3. A student writes (sin²x + 4) / (sin x + 2) = sin x + 2. Test with x = 30° and decide if it is valid.
Show answer
At x = 30°: the left side is (0.25 + 4) / (0.5 + 2) = 4.25 / 2.5 = 1.7, but sin x + 2 = 2.5. Not valid, because sin²x + 4 does not factorise as (sin x + 2)(sin x + 2).
Where this leads next
With legal cancelling secure, try the whole module in the mixed practice set, and log any slips in the mistake log and retest queue. The non-calculator working trainer gives extra algebra practice.
Students who lose marks to a tidy but wrong line often need someone to read their working closely, which is part of online one-to-one Additional Mathematics tuition.