Trigonometric Identities and Symmetries

Trigonometric identities are equations that hold for all angles where both sides are defined, and the symmetries of the unit circle give simple rules such as sin(−x) = −sin x, cos(−x) = cos x and sin(π − x) = sin x. In brief: the key identities are sin²x + cos²x = 1, the sum formulas sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b − sin a sin b, and the double-angle formulas sin 2x = 2 sin x cos x and cos 2x = cos²x − sin²x.

Identities let you simplify expressions, solve equations and compute exact values such as sin 75° or cos 15°. This guide lists the identities in groups: symmetry rules, the Pythagorean identities, the sum and difference formulas, double and half angles and product-to-sum formulas, with derivations of the central ones and worked examples.

What are the symmetry rules?

On the unit circle, the point at angle θ is (cos θ, sin θ). Reflecting the point across the axes or the origin gives the following rules.

• Negative angle (reflection in the x-axis): sin(−x) = −sin x, cos(−x) = cos x, tan(−x) = −tan x.
• Supplementary angles (reflection in the y-axis): sin(π − x) = sin x, cos(π − x) = −cos x, tan(π − x) = −tan x.
• Half-turn (reflection in the origin): sin(π + x) = −sin x, cos(π + x) = −cos x, tan(π + x) = tan x.
• Cofunctions (reflection in y = x): sin(π/2 − x) = cos x, cos(π/2 − x) = sin x.
• Periodicity: sin(x + 2π) = sin x and cos(x + 2π) = cos x; tan(x + π) = tan x.

Example: sin 150° = sin(180° − 30°) = sin 30° = 1/2. And cos 225° = cos(180° + 45°) = −cos 45° = −√2/2. And cos(−60°) = cos 60° = 1/2.

What are the Pythagorean identities?

Since the point (cos x, sin x) lies on the unit circle, sin²x + cos²x = 1. Dividing by cos²x gives 1 + tan²x = 1/cos²x, and dividing by sin²x gives 1 + cot²x = 1/sin²x. They hold for all x where the quantities are defined.

The forms sin²x = 1 − cos²x and cos²x = 1 − sin²x are used to convert between sine and cosine. When taking a root, the sign depends on the quadrant: sin x = ±√(1 − cos²x).

What are the sum and difference formulas?

• sin(a + b) = sin a cos b + cos a sin b
• sin(a − b) = sin a cos b − cos a sin b
• cos(a + b) = cos a cos b − sin a sin b
• cos(a − b) = cos a cos b + sin a sin b
• tan(a + b) = (tan a + tan b)/(1 − tan a tan b)

A derivation of cos(a − b): take two points on the unit circle at angles a and b, P(cos a, sin a) and Q(cos b, sin b). The squared distance is (cos a − cos b)² + (sin a − sin b)² = 2 − 2(cos a cos b + sin a sin b). By rotating the figure so that Q is at (1, 0), the distance equals the distance between (cos(a − b), sin(a − b)) and (1, 0), which is 2 − 2 cos(a − b). Equating the two gives cos(a − b) = cos a cos b + sin a sin b. The other formulas follow by substituting −b and using cofunctions.

Example 1: exact value of sin 75°

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4 ≈ 0.9659.

Example 2: cos 15°

cos 15° = cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30° = (√6 + √2)/4 ≈ 0.9659. This equals sin 75°, as expected from the cofunction rule.

Example 3: tan 105°

tan(60° + 45°) = (√3 + 1)/(1 − √3) = −(√3 + 1)²/2 = −(4 + 2√3)/2 = −(2 + √3) ≈ −3.732.

What are the double-angle and half-angle formulas?

Setting a = b = x in the sum formulas gives:

• sin 2x = 2 sin x cos x
• cos 2x = cos²x − sin²x = 2 cos²x − 1 = 1 − 2 sin²x
• tan 2x = 2 tan x/(1 − tan²x)

Solving the cosine forms for the squares gives the power-reduction (half-angle) formulas: sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2. Replacing x by x/2 gives sin(x/2) = ±√((1 − cos x)/2) and cos(x/2) = ±√((1 + cos x)/2).

Example 4

If sin x = 3/5 with x in quadrant I, then cos x = 4/5. So sin 2x = 2 · (3/5)(4/5) = 24/25 and cos 2x = 16/25 − 9/25 = 7/25. Check: (24/25)² + (7/25)² = (576 + 49)/625 = 1 ✓.

Example 5: sin 15° from a half-angle

sin 15° = √((1 − cos 30°)/2) = √((1 − √3/2)/2) = √(2 − √3)/2 ≈ 0.2588.

How do you prove an identity?

1. Choose the more complicated side and work on it.
2. Rewrite in sines and cosines; combine fractions with a common denominator.
3. Use the Pythagorean and other identities to simplify.
4. Stop when you reach the other side.

Example 6: prove (1 − cos 2x)/sin 2x = tan x

Use cos 2x = 1 − 2 sin²x and sin 2x = 2 sin x cos x: (2 sin²x)/(2 sin x cos x) = sin x/cos x = tan x ✓.

Example 7: prove sin⁴x − cos⁴x = sin²x − cos²x

Difference of squares: (sin²x − cos²x)(sin²x + cos²x) = (sin²x − cos²x) · 1 ✓.

Summary table of identities

Where do the identities come from?

Ptolemy’s theorem on cyclic quadrilaterals, in the second century, gave the equivalents of the sum and difference formulas for chords. Indian and Islamic mathematicians developed the formulas for sines, and Viète and others in the sixteenth century wrote multiple-angle formulas. Euler’s formula e^ix = cos x + i sin x, from the eighteenth century, gives a short derivation of all these identities.

Where are they used today?

Identities simplify integrals such as ∫ sin²x dx = x/2 − sin 2x/4 + C. In physics, the sum formulas describe interference of waves and the beat frequencies of two close tones. In signal processing, product-to-sum formulas explain modulation, and in navigation they are used to compute the position after a rotation.

Which mistakes are common?

Distributing sine over a sum

sin(a + b) is not sin a + sin b. Check: sin 90° = 1, but sin 60° + sin 30° = 1.366.

Mixing up the signs in the cosine formulas

cos(a + b) has a minus between the products, and cos(a − b) has a plus.

Writing sin 2x = 2 sin x

The correct formula is sin 2x = 2 sin x cos x. For x = 30°: sin 60° = 0.866, but 2 sin 30° = 1.

Ignoring the quadrant when taking roots

The half-angle formulas have ±, and the sign depends on the quadrant of x/2.