An excluded value is an angle where the identity cannot be used because something in it is undefined. Most of the time that means a denominator equal to zero. Some Additional Mathematics questions ask for these angles in a stated range, and careful students note them even when the question does not ask.
This lesson follows simplifying before substitution and belongs to trigonometric identities.
Where do excluded values come from?
A trig function can be undefined in four ways.
- tan x and sec x are undefined when cos x = 0.
- cot x and cosec x are undefined when sin x = 0.
- Any fraction is undefined when its denominator is zero, such as 1 − cos x or sin x + cos x.
- Cancelling a factor assumes that factor is not zero, so any cancelled factor adds a restriction.
How do you find them?
- Look at the original expression, not the simplified one. Write down each denominator and each function that is undefined somewhere.
- Set each one equal to zero and solve in the given range.
- Do this for both sides. Take the union of the results.
- List the angles in the stated range, in degrees or radians as the question uses.
Worked example
Prove that (1 − cos x) / sin x ≡ sin x / (1 + cos x), and state the values of x in 0° ≤ x ≤ 360° for which it is valid.
Proof from the left side: multiply the numerator and denominator by (1 + cos x):
(1 − cos x)(1 + cos x) / [sin x (1 + cos x)] = (1 − cos²x) / [sin x (1 + cos x)] = sin²x / [sin x (1 + cos x)] = sin x / (1 + cos x).
Left side undefined: sin x = 0, so x = 0°, 180°, 360°.
Right side undefined: 1 + cos x = 0, so cos x = −1, so x = 180°.
Union: the identity is valid for all x in the range except x = 0°, 180° and 360°.
The right side is defined at 0° and 360° (it equals 0 there), but the left side is 0/0, so those angles are excluded from the identity. A check at x = 90°: left is (1 − 0) / 1 = 1, and right is 1 / (1 + 0) = 1.
The mistake to watch for
A common slip is to state restrictions only from the simplified form, or to state none at all.
Mistaken working: (sin x cos x) / cos x = sin x, so it is true for all x.
The student cancelled cos x and forgot that the original expression is not defined when cos x = 0.
The correction is to look at the original fraction.
It is undefined when cos x = 0, so x = 90° and 270° in 0° ≤ x ≤ 360°. The identity (sin x cos x) / cos x = sin x holds everywhere else. At x = 90° the left side is 0/0, while the right side is 1.
Check yourself
1. State the values of x in 0° ≤ x ≤ 360° for which tan x is undefined.
Show answer
tan x = sin x / cos x is undefined when cos x = 0, so x = 90° and 270°.
2. State the values of x in 0° ≤ x ≤ 360° for which 1 / (sin x − cos x) is undefined.
Show answer
The denominator is zero when sin x = cos x, which means tan x = 1. So x = 45° and 225°. At 225° both sin and cos equal −0.707, so they are equal.
3. State the values of x in 0° ≤ x ≤ 360° for which cosec x + cot x is undefined.
Show answer
Both cosec x and cot x have sin x in the denominator, so the expression is undefined when sin x = 0: x = 0°, 180° and 360°.
Where this leads next
Next, diagnose an invalid cancellation, because most illegal cancelling also ignores restrictions. The triangle and bearings reasoning board helps you picture which ratios exist at an angle, and the non-calculator working trainer supports the algebra.
Careful restriction statements are a habit, and our teachers build it in online one-to-one Additional Mathematics tuition.