Grade by Grade
D. Trigonometric Functions · Trigonometric Ratios, Identities and Laws

Trigonometric Identities: Pythagorean, Quotient and Reciprocal

D1.5About 70 minutes
Learning goals
  • I can explain the difference between an identity and an equation.
  • I can develop the quotient identity and the Pythagorean identity from x, y and r.
  • I can prove simple identities by simplifying each side separately.
  • I can use the Pythagorean identity to find one ratio from another.
  • I can show that an equation is not an identity by finding a counterexample.

Words to know

identity
An equation that is true for every value of the variable for which both sides are defined, such as sin²θ + cos²θ = 1.
quotient identity
tan θ = sin θ/cos θ (for cos θ ≠ 0).
Pythagorean identity
sin²θ + cos²θ = 1.
reciprocal identities
csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
counterexample
One value that makes a statement false. One counterexample is enough to show an equation is not an identity.
L.S. and R.S.
The left side and the right side of an equation.

Equations that are always true

The equation 2x + 3 = 7 is true only when x = 2. But the equation 2(x + 3) = 2x + 6 is true for every x. The second kind is called an identity. You met identities in the unit Functions and Equivalent Expressions, when you checked whether two expressions are equivalent.

Trigonometry has its own identities. They let you rewrite one expression as another, simpler one, which is a key skill for Grade 12 Advanced Functions (MHF4U).

The quotient identity

With a point (x, y) on the terminal arm and distance r:

sin θ ÷ cos θ = (y/r) ÷ (x/r) = (y/r) × (r/x) = y/x = tan θ

So tan θ = sin θ/cos θ, whenever cos θ ≠ 0. Flipping both sides gives cot θ = cos θ/sin θ.

The Pythagorean identity

The point (x, y) is r units from the origin, so x² + y² = r². Divide every term by r²:

x²/r² + y²/r² = 1, which is (cos θ)² + (sin θ)² = 1.

We write sin²θ + cos²θ = 1. It is true for every angle, because it is really the Pythagorean theorem. On the unit circle you can see it directly: the triangle under the point (cos θ, sin θ) has legs cos θ and sin θ and hypotenuse 1.

θ1cos θsin θP(cos θ, sin θ)Pythagorean theorem in the triangle:(cos θ)² + (sin θ)² = 1²so sin²θ + cos²θ = 1Rearranged:sin²θ = 1 − cos²θcos²θ = 1 − sin²θ
On the unit circle the legs of the triangle are cos θ and sin θ and the hypotenuse is 1. The Pythagorean theorem gives sin²θ + cos²θ = 1.

The identities you need for this course

  • Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
  • Quotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
  • Pythagorean: sin²θ + cos²θ = 1, so also sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ

Each one holds wherever both sides are defined (for example, tan θ = sin θ/cos θ needs cos θ ≠ 0).

Worked example

Example 1: Checking an identity with one angle

Verify sin²θ + cos²θ = 1 for θ = 120°.

  1. sin 120° = √3/2, so sin²120° = 3/4.
  2. cos 120° = −1/2, so cos²120° = 1/4.
  3. 3/4 + 1/4 = 1. ✓

It works for 120°. But checking one angle does not prove an identity. A proof must work for every angle.

How to prove an identity

Treat the two sides as separate expressions. Rewrite one side, or each side, step by step using known identities and algebra, until both sides are the same expression.

  • Write L.S. = and R.S. = and work on them separately.
  • Never move terms across the equals sign, and never do the same thing to both sides. Those moves assume the identity is already true, which is what you are trying to show.
  • It often helps to start with the more complicated side and to rewrite everything in terms of sin θ and cos θ.
  • End with a statement like "L.S. = R.S., so the identity is true."
Worked example

Example 2: A one-step proof

Prove tan θ cos θ = sin θ.

  1. L.S. = tan θ cos θ = (sin θ/cos θ) × cos θ (quotient identity).
  2. Cancel cos θ: L.S. = sin θ.
  3. R.S. = sin θ.

L.S. = R.S., so tan θ cos θ = sin θ is an identity (for cos θ ≠ 0).

L.S. = tan θ cos θ= (sin θ / cos θ) × cos θ= sin θR.S. = sin θ(nothing to simplify)L.S. = R.S., so the identity is proven
Lay out a proof in two separate columns. Simplify each side on its own until they match.
Worked example

Example 3: Using the Pythagorean identity

Prove (1 − sin²θ)/cos θ = cos θ.

  1. L.S. = (1 − sin²θ)/cos θ.
  2. Replace 1 − sin²θ with cos²θ (Pythagorean identity): L.S. = cos²θ/cos θ.
  3. Simplify: L.S. = cos θ.
  4. R.S. = cos θ.

L.S. = R.S., so the identity is proven.

Worked example

Example 4: Using reciprocal identities

Prove sec θ cot θ = csc θ.

  1. L.S. = sec θ cot θ = (1/cos θ) × (cos θ/sin θ).
  2. Cancel cos θ: L.S. = 1/sin θ = csc θ.
  3. R.S. = csc θ.

L.S. = R.S. ✓

Worked example

Example 5: Finding one ratio from another

sin θ = 2/3 and θ is in quadrant II. Find the exact values of cos θ and tan θ.

  1. cos²θ = 1 − sin²θ = 1 − 4/9 = 5/9.
  2. cos θ = ±√5/3. In quadrant II cosine is negative, so cos θ = −√5/3.
  3. tan θ = sin θ/cos θ = (2/3) ÷ (−√5/3) = −2/√5 = −2√5/5.

cos θ = −√5/3, tan θ = −2√5/5

Worked example

Example 6: Not an identity

Is sin θ + cos θ = 1 an identity?

  1. Test θ = 0°: 0 + 1 = 1. True. Test θ = 90°: 1 + 0 = 1. True.
  2. Test θ = 45°: √2/2 + √2/2 = √2 ≈ 1.41. Not 1.
  3. One counterexample is enough.

No. It is true for some angles (it is an equation with solutions), but θ = 45° is a counterexample, so it is not an identity.

Just for fun

Why don't identities ever get nervous before a test?

They know they'll be true no matter what (well, wherever they're defined).

Hands-on activity

Identity or not? A calculator investigation

You need: a calculator in degree mode, paper, and a die or two (optional, for picking random angles).

  1. Pick five angles between 0° and 360°. For random angles, roll two dice, multiply the numbers and then multiply by 10 (for example, 4 and 5 give 200°).
  2. Make a table with columns θ, sin²θ + cos²θ, sin θ + cos θ, and tan θ − sin θ/cos θ.
  3. Fill in each column for your five angles. (Type sin²θ as (sin θ)².)
  4. Which columns give the same number every time?

What to look for: sin²θ + cos²θ should be 1 every time, and tan θ − sin θ/cos θ should be 0 every time (unless your angle is 90° or 270°, where tan is undefined). sin θ + cos θ changes from angle to angle, so it is not an identity. Remember: the table is evidence, not a proof. The proofs are in this lesson.

Your turn

1

Verify that sin²30° + cos²30° = 1.

2

Prove sin θ csc θ = 1.

3

Prove cot θ sin θ = cos θ.

4

Prove sin θ/tan θ = cos θ.

5

cos θ = −5/13 and θ is in quadrant III. Find sin θ and tan θ.

6

Use θ = 30° to show that sin θ + cos θ = 1 is not an identity.

7

Prove 1 − cos²θ = sin²θ.

More practice: warm-up

1

Write the quotient identity for tan θ.

2

What does sin²θ mean?

3

sin²θ = 0.36. What is cos²θ?

4

Which ratio is the reciprocal of cos θ?

5

Write cot θ in terms of sin θ and cos θ.

More practice: on your own

1

Prove tan θ csc θ = sec θ.

2

Prove (1 − cos²θ)/sin θ = sin θ.

3

Prove sin θ cos θ tan θ = 1 − cos²θ.

4

sin θ = −0.6 and θ is in quadrant IV. Find cos θ and tan θ.

5

Prove tan θ/sec θ = sin θ.

Challenge

1

Prove 1 + tan²θ = sec²θ.

2

Prove 1 + cot²θ = csc²θ.

3

tan θ = 2 and θ is in quadrant III. Find the exact values of sin θ and cos θ.

Check your understanding

Quiz

1.

An identity is:

2.

Which is the quotient identity?

3.

sin²θ + cos²θ equals:

4.

cos θ = 0.6 and θ is in quadrant IV. What is sin θ?

5.

What is the best way to prove an identity?

6.

Which equation is NOT an identity?

For parents and teachers

Big idea (D1.5): an identity is true for all angles (where defined). Students prove simple identities using the Pythagorean identity sin²θ + cos²θ = 1, the quotient identity tan θ = sin θ/cos θ, and the reciprocal identities.

How to help: Look at the layout of a proof. There should be two separate columns or blocks (L.S. and R.S.), and no step should move a term across the equals sign. If your teen gets stuck, the usual hint is "write everything in sin and cos".

Common difficulties:

  • Treating a proof like an equation to solve (moving terms, cross-multiplying).
  • Believing that one angle that works proves the identity. One angle can only disprove (a counterexample).
  • Forgetting the ± sign when taking a square root, and then choosing the sign with CAST.