Grade by Grade
B. Trigonometric Functions · Trigonometric Identities and Equations

Proving Identities with Compound and Double Angle Formulas

B3.3 B3.2About 75 minutes
Learning goals
  • I can prove identities that need compound angle formulas, such as sin(x + y) sin(x − y) = sin²x − sin²y.
  • I can prove identities with double angles, choosing the best form of cos 2x.
  • I can combine compound or double angle formulas with the Pythagorean, quotient and reciprocal identities in one proof.
  • I can use a counterexample or graphing technology to check a claim before I try to prove it.

Words to know

expand
Replace a compound or double angle with its formula, such as sin 2x → 2 sin x cos x.
contract
Recognize a formula pattern and replace it with a single ratio, such as cos²x − sin²x → cos 2x.
verify with technology
Graph both sides of a claimed identity; if the graphs coincide over many periods, the claim is very likely an identity, which a proof then confirms.

More tools, same rules

The rules from the last lesson do not change: work on each side separately, never move terms across the equals sign, and justify every step. What is new is the toolbox. When you see an angle like 2x, x + y or x − π/3, your first move is usually to expand it with a compound or double angle formula, so that every term uses the same angle. Sometimes the reverse helps: contract a pattern like 2 sin x cos x into sin 2x.

The curriculum asks you to prove identities "using a variety of relationships". The proofs in this lesson mix all the identities you know.

If you see ...... replace it with1 + cos 2x2cos²x1 − cos 2x2sin²xsin 2x2 sin x cos xcos 2x, rest in sin x1 − 2sin²xcos 2x, rest in cos x2cos²x − 1
A guide to the double angle substitutions. The forms 1 + cos 2x = 2cos²x and 1 − cos 2x = 2sin²x are especially useful because the 1 cancels.
Worked example

Example 1: A compound angle product

Prove sin(x + y) sin(x − y) = sin²x − sin²y.

  1. L.S. = (sin x cos y + cos x sin y)(sin x cos y − cos x sin y).
  2. This is a difference of squares: = sin²x cos²y − cos²x sin²y.
  3. The right side has only sines, so replace the cosines: = sin²x(1 − sin²y) − (1 − sin²x) sin²y.
  4. = sin²x − sin²x sin²y − sin²y + sin²x sin²y = sin²x − sin²y = R.S.

L.S. = R.S. ✓

Worked example

Example 2: Choosing the form of cos 2x

Prove sin 2x/(1 + cos 2x) = tan x.

  1. The denominator is 1 + cos 2x, so use cos 2x = 2cos²x − 1 to cancel the 1.
  2. L.S. = 2 sin x cos x/(1 + 2cos²x − 1) = 2 sin x cos x/(2cos²x).
  3. = sin x/cos x = tan x = R.S.

L.S. = R.S. ✓ (where cos x ≠ 0 and 1 + cos 2x ≠ 0, that is, x ≠ π/2 + kπ).

Worked example

Example 3: Expanding a square

Prove (sin x + cos x)² = 1 + sin 2x.

  1. L.S. = sin²x + 2 sin x cos x + cos²x.
  2. Regroup: = (sin²x + cos²x) + 2 sin x cos x = 1 + 2 sin x cos x.
  3. Contract the double angle: = 1 + sin 2x = R.S.

L.S. = R.S. ✓ for every real x.

xπ/2π3π/22π21π/4: both sides = 23π/4: both sides = 0y = (sin x + cos x)² and y = 1 + sin 2x
Graphing technology shows y = (sin x + cos x)² and y = 1 + sin 2x as one curve. Check two points by hand: at π/4 both sides are 2; at 3π/4 both are 0.
Worked example

Example 4: Factoring with a double angle

Prove cos⁴x − sin⁴x = cos 2x.

  1. L.S. = (cos²x − sin²x)(cos²x + sin²x), a difference of squares.
  2. = (cos²x − sin²x)(1).
  3. = cos 2x = R.S.

L.S. = R.S. ✓

Worked example

Example 5: Special angles inside a proof

Prove cos(x + π/3) + cos(x − π/3) = cos x.

  1. L.S. = (cos x cos(π/3) − sin x sin(π/3)) + (cos x cos(π/3) + sin x sin(π/3)).
  2. The sine terms cancel: = 2 cos x cos(π/3).
  3. = 2 cos x (1/2) = cos x = R.S.

L.S. = R.S. ✓

Worked example

Example 6: Reciprocals and double angles

Prove csc 2x + cot 2x = cot x.

  1. L.S. = 1/sin 2x + cos 2x/sin 2x = (1 + cos 2x)/sin 2x.
  2. = 2cos²x/(2 sin x cos x).
  3. = cos x/sin x = cot x = R.S.

L.S. = R.S. ✓ (for x ≠ kπ/2).

When you get stuck

  • Make all the angles the same (expand 2x, x + y and so on).
  • Make all the functions sine and cosine.
  • Look at the target side: if it has only sines, replace cos² with 1 − sin²; if it has tan x, aim for sin x/cos x.
  • Look for patterns to contract: 2 sin x cos x, cos²x − sin²x, a difference of squares.
  • Before a long attempt, test one value. A failed test means there is nothing to prove.
Just for fun

cos 2x is the only double angle formula with three different outfits: cos²x − sin²x, 2cos²x − 1 and 1 − 2sin²x. Choosing the right one for the occasion is half the proof.

Hands-on activity

Check an identity with exact values

You need: paper, a pencil, your table of exact values for π/6, π/4, π/3 and π/2, and (optional) a free graphing app such as Desmos.

Step 1. Make a table with x = 0, π/6, π/4, π/3 and π/2. For each x, work out the exact value of (sin x + cos x)² and of 1 + sin 2x, without a calculator.

Step 2. Compare the columns.

Step 3 (optional). Graph both sides in radian mode and watch them coincide. Then graph y = (sin x + cos x)² and y = 1 + cos 2x and watch them separate.

What to look for: both columns should read 1, 1 + √3/2, 2, 1 + √3/2, 1. For π/6: (1/2 + √3/2)² = (1 + 2√3 + 3)/4 = 1 + √3/2, and 1 + sin(π/3) = 1 + √3/2. The table supports the identity; Example 3 proves it.

Your turn

1

Prove (cos x − sin x)² = 1 − sin 2x.

2

Prove (1 − cos 2x)/(1 + cos 2x) = tan²x.

3

Prove sin 2x/sin x − cos 2x/cos x = sec x.

4

Prove sin(π/4 + x) − sin(π/4 − x) = √2 sin x.

5

Prove tan x + cot x = 2 csc 2x.

6

Is sin 2x cos x = sin 3x an identity? Justify.

More practice: warm-up

1

Write 1 + cos 2x as a single term.

2

Write 1 − cos 2x as a single term.

3

Contract 2 sin 3x cos 3x.

4

Contract cos²(x/2) − sin²(x/2).

5

Simplify sin 2x/(2 cos x).

More practice: on your own

1

Prove sin(x + y)/sin(x − y) = (tan x + tan y)/(tan x − tan y).

2

Prove cos 2x = (1 − tan²x)/(1 + tan²x).

3

Prove sin 2x = 2 tan x/(1 + tan²x).

4

Prove (sin 2x + sin x)/(cos 2x + cos x + 1) = tan x.

5

Prove 2 sin(x + π/6) − √3 sin x = cos x.

Challenge

1

Prove sin 3x/sin x − cos 3x/cos x = 2.

2

Prove tan(x + π/4) − tan(x − π/4) = 2 sec 2x.

3

Prove cos⁴x = 3/8 + (1/2)cos 2x + (1/8)cos 4x. (Formulas like this are used in calculus.)

Check your understanding

Quiz

1.

In a proof, what should 1 + cos 2x usually be replaced with?

2.

Simplify sin 2x/(1 + cos 2x).

3.

Which expression equals (sin x + cos x)²?

4.

Which expression equals cos⁴x − sin⁴x?

5.

Simplify cos(x + π/3) + cos(x − π/3).

6.

The graphs of both sides of a claimed identity coincide on your screen. What should you do next?

For parents and teachers

Big idea (B3.3, B3.2): proofs now use every identity in the unit: compound angle, double angle, Pythagorean, quotient and reciprocal. A good first move is to expand any compound or double angle so all angles match, then rewrite in sine and cosine.

How to help: Have your teen keep a one-page "identity sheet" with every formula from this unit. When they are stuck, ask: "Are all the angles the same? Is everything in sine and cosine? Which form of cos 2x fits?" Checking both sides on a free graphing app (radian mode) before proving is a good habit.

Common difficulties:

  • Using the wrong form of cos 2x and getting tangled.
  • Sign errors when expanding compound angles.
  • Not recognizing a pattern to contract (2 sin x cos x = sin 2x).
  • Forgetting that a coinciding graph is evidence, not proof.