B. Trigonometric Functions · Trigonometric Identities and Equations

Identities, Equations and Counterexamples

B3.3 B3.1About 65 minutes

Learning goals

I can explain the difference between a trigonometric identity and an equation that is true only for some values.
I can use a single counterexample to show that an equation is not an identity, choosing test values wisely.
I can state the restrictions (values excluded from the domain) for both sides of an identity.
I can use graphs and tables from technology to test whether an equation might be an identity.

Words to know

identity
An equation that is true for every value of the variable for which both sides are defined, such as sin²x + cos²x = 1.

conditional equation
An equation that is true only for certain values of the variable, such as sin x = 1/2. You solve it to find those values.

counterexample
One specific value that makes the two sides of a claimed identity different. One is enough to prove the claim false.

restriction
A value excluded from the domain because some part of the equation is undefined there, such as x ≠ π/2 + kπ when tan x appears.

domain
The set of values of the variable for which an expression is defined.

Always, sometimes, or never?

Look at three equations:

sin²x + cos²x = 1 is true for every x.
sin x = 1/2 is true for some x (π/6, 5π/6, and those plus multiples of 2π).
sin x = 2 is true for no x.

The first kind is an identity. The curriculum puts it this way: identities are equations that are true for every value in the domain. The second kind is a conditional equation; you solve it, which is the topic of lessons 10 to 14. This lesson is about telling them apart, and about one powerful idea: to show that an equation is not an identity, you need just one counterexample.

Your identity toolkit so far

Remember from Grade 11 (MCR3U, Trigonometric Ratios, Identities and Laws):

Reciprocal: csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x
Quotient: tan x = sin x/cos x, cot x = cos x/sin x
Pythagorean: sin²x + cos²x = 1, 1 + tan²x = sec²x, 1 + cot²x = csc²x

New this unit: cofunction and symmetry identities (lessons 1 and 2), the compound angle formulas (lessons 3 to 5) and the double angle formulas (lesson 6).

How to test a claimed identity

Try a value. Substitute a number for x on both sides. If the sides differ, you are done: it is not an identity, and that value is your counterexample.
If the sides agree, try more values. Agreement at a few values is evidence, not proof. An equation can agree at many values and still fail somewhere.
Graph both sides with technology in radian mode. If the graphs coincide everywhere you look, the equation is very likely an identity. If they separate anywhere, it is not.
Prove it. Only a proof (lessons 8 and 9) shows that an identity holds for every x in the domain.

Left: y = cos 2x and y = 1 − 2sin²x give one curve, the sign of an identity. Right: y = cos 2x and y = 2 cos x − 1 meet only at the marked points, so cos 2x = 2 cos x − 1 is a conditional equation, not an identity.

Choose test values carefully

Bad test values make false claims look true:

x = 0 makes sin x = 0 and tan x = 0, so many terms vanish. (sin 2x = 2 sin x is true at x = 0 but false in general.)
x = π/4 makes sin x = cos x, so any claim that mixes up sine and cosine passes.
Multiples of π/2 make one ratio 0 or undefined.

Good choices: π/6, π/3, or a "random" number such as x = 1 or x = 2 on a calculator in radian mode.

Worked example

Example 1: A counterexample

Show that sin(x + y) = sin x + sin y is not an identity.

Choose x = y = π/2.
Left side: sin(π/2 + π/2) = sin π = 0.
Right side: sin(π/2) + sin(π/2) = 1 + 1 = 2.
0 ≠ 2, so the equation fails at this value.

Not an identity. x = y = π/2 is a counterexample. (The true formula is sin x cos y + cos x sin y.)

Worked example

Example 2: Passing one test is not enough

Is cos 2x = 2 cos x − 1 an identity?

Test x = 0: left = cos 0 = 1, right = 2(1) − 1 = 1. They agree... but that proves nothing.
Test x = π/3: left = cos(2π/3) = −1/2, right = 2(1/2) − 1 = 0.
−1/2 ≠ 0, so x = π/3 is a counterexample.
The graph (right panel above) shows the curves meet at x = 0, π/2, 3π/2 and 2π only. It is a conditional equation that happens to be true at those four values.

Not an identity.

Worked example

Example 3: Restrictions

Explain why (1 − cos²x)/sin x = sin x is an identity, and state its restrictions.

The left side is undefined when sin x = 0, that is, when x = kπ for any integer k. The right side is defined everywhere.
For every other x: (1 − cos²x)/sin x = sin²x/sin x = sin x.
So the two sides agree for every x in the domain of both sides.

It is an identity for x ≠ kπ (k an integer).

Worked example

Example 4: Test, then prove

Test sec²x − tan²x = 1 at x = π/3, then prove it.

Test: sec(π/3) = 2, so sec² = 4. tan(π/3) = √3, so tan² = 3. 4 − 3 = 1 ✓.
This supports the claim but does not prove it.
Proof: from the Pythagorean identity 1 + tan²x = sec²x, subtract tan²x from both sides to get sec²x − tan²x = 1, for every x where tan x and sec x are defined (x ≠ π/2 + kπ).

It is an identity for x ≠ π/2 + kπ.

The graph of y = tan x cos x is the sine curve with holes where tan x is undefined. So tan x cos x = sin x is an identity only on the domain x ≠ π/2 + kπ. Graphing technology usually hides these holes, so you must find restrictions yourself.

Just for fun

One counterexample is enough to sink a claim. For centuries Europeans said "all swans are white", until people in Europe learned of black swans, which are native to Australia. One black swan, and the claim was done.

Hands-on activity

Identity or not? A calculator sort

You need: a calculator in radian mode, paper and a pencil.

Write each equation on its own slip of paper. Test each one at x = 1 and x = 2, then sort the slips into two piles: "definitely not an identity" and "might be an identity".

sin²x + cos²x = 1
sin 2x = 2 sin x
cos 2x = 1 − 2sin²x
tan x = sin x/cos x
cos(x + 1) = cos x + cos 1
sin(π − x) = sin x
(sin x + cos x)² = 1
sec x − cos x = sin x tan x

Answers: definitely not identities: 2 (at x = 1, 0.9093 vs 1.6829), 5 (at x = 1, −0.4161 vs 1.0806) and 7 (at x = 1, 1.9093 vs 1). The others agree at both values (for 8, both sides are about 1.3105 at x = 1 and −1.9869 at x = 2), and all five are true identities: 1 and 4 from Grade 11, 3 from lesson 6, 6 from lesson 2, and 8 is proved in the practice below.

Your turn

Is tan x = sin x cos x an identity? Justify.

Is (sin x + cos x)² = sin²x + cos²x an identity? Justify.

State the restrictions for tan x/sin x = sec x.

Show that cos 2x = cos²x is not an identity.

Test 1 + cot²x = csc²x at x = π/6. Does your test prove the identity?

Is sin x = cos x an identity or a conditional equation? Give one value where it is true and one where it is false.

More practice: warm-up

What is a trigonometric identity?

Is sin²x + cos²x = 1 an identity?

Is sin x = 1/2 an identity?

How many counterexamples do you need to show an equation is not an identity?

If both sides agree at x = π/6, π/4 and π/3, is the equation definitely an identity?

More practice: on your own

Show that sin(π/2 + x) = sin(π/2) + sin x is not an identity.

Explain why x = 0 is a poor test for sin 2x = 2 sin x, then find a counterexample.

State the restrictions for sin x/(1 + cos x) = (1 − cos x)/sin x.

cos 2x = 2 cos x − 1 is not an identity. Find every x in [0, 2π] where it is true.

A graph shows y = sec x − cos x and y = sin x tan x on top of each other. Prove that sec x − cos x = sin x tan x and state the restrictions.

Challenge

A student checks sin 3x = 3 sin x at x = 0, π and 2π, finds both sides equal each time, and concludes it is an identity. Find the flaw and a counterexample.

Is (tan x + 1)/(tan x − 1) = (1 + cot x)/(1 − cot x) an identity? Justify and give restrictions.

Is sin x = √(1 − cos²x) an identity? What about |sin x| = √(1 − cos²x)?

Check your understanding

Quiz

What makes an equation a trigonometric identity?

Which value of x shows that sin 2x = 2 sin x is not an identity?

What is the restriction on tan x cos x = sin x?

Both sides of an equation agree at x = π/6, π/4 and π/3. What can you conclude?

Which describes sin x = cos x?

Which equation is an identity?

For parents and teachers

Big idea (B3.3): an identity is true for every value in its domain; a conditional equation is true only for some values. One counterexample is enough to show an equation is not an identity, but checking a few values can never prove that it is.

How to help: Play "identity or not?" Write an equation, and have your teen test it on a calculator in radian mode at x = 1 and x = 2. If it fails, they have a counterexample. If it passes, ask, "Does that prove it?" (No: that needs a proof, which comes in the next lessons.)

Common difficulties:

Testing only at x = 0 or x = π/4, where many false claims happen to work.
Calling an equation an identity after checking one or two values.
Ignoring restrictions: tan x, sec x, csc x, cot x and fractions all exclude some values.
Mixing up "solve" (find values) and "prove" (show it is always true).