Trigonometric Identities Made Simple: Angle Sum and Difference Identities

Why angle sum and difference identities matter

Angle sum/difference identities let you break a trig function of a combined angle (like x+30° or a-b) into pieces involving the single angles. This is especially useful when (1) you want exact values for angles that are not “basic” by themselves (like 75° or 15°), and (2) you want to rewrite expressions so everything is in terms of sin x and cos x (or tan x) without needing product-to-sum or sum-to-product conversions.

Geometric setup (unit-circle / rotation viewpoint)

Think of a point on the unit circle at angle a as the vector (cos a, sin a). Rotating any vector by angle b is a geometric action that can be captured algebraically by a rotation rule. If you rotate the vector (x, y) by b, the new coordinates become:

(x', y') = (x cos b - y sin b, x sin b + y cos b)

This rule can be justified by projecting the rotated vector onto the axes (unit-circle reasoning) and using right-triangle relationships for the components along the new directions.

Deriving sin(a+b) and cos(a+b) step by step

Step 1: Start with the unit vector at angle a

On the unit circle, the point at angle a is:

(cos a, sin a)

Step 2: Rotate it by b

Apply the rotation rule with x = cos a and y = sin a:

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x' = (cos a)cos b - (sin a)sin b  y' = (cos a)sin b + (sin a)cos b

Step 3: Recognize what the rotated point represents

Rotating the point at angle a by b lands you at angle a+b. So the new coordinates must also equal:

(cos(a+b), sin(a+b))

Step 4: Match coordinates to get the identities

Match the x-coordinates and y-coordinates:

• cos(a+b) = cos a cos b - sin a sin b
• sin(a+b) = sin a cos b + cos a sin b

Difference identities from the sum identities

Use the same formulas but replace b with -b. On the unit circle, cos(-b)=cos b and sin(-b)=-sin b. Substitute into the sum formulas:

• cos(a-b) = cos a cos b + sin a sin b
• sin(a-b) = sin a cos b - cos a sin b

Deriving tan(a±b) using algebra (from sine and cosine)

Start from tan(θ)=sin(θ)/cos(θ). Then:

tan(a+b) = sin(a+b)/cos(a+b)

Substitute the identities you just derived:

tan(a+b) = (sin a cos b + cos a sin b) / (cos a cos b - sin a sin b)

Now divide numerator and denominator by cos a cos b (a clean algebra move that reveals tangents):

tan(a+b) = ( (sin a/cos a) + (sin b/cos b) ) / ( 1 - (sin a/cos a)(sin b/cos b) )

So:

• tan(a+b) = (tan a + tan b) / (1 - tan a tan b)

Similarly, replacing b with -b gives:

• tan(a-b) = (tan a - tan b) / (1 + tan a tan b)

Practical caution: these formulas require the relevant cosines to be nonzero (so the tangents exist), and the denominators 1 - tan a tan b or 1 + tan a tan b must not be zero.

Quick reference table (sum and difference)

How these identities produce exact values for special angles

The strategy is to express a “nonstandard” angle as a sum or difference of angles with known exact trig values (often multiples of 30° and 45°). Then apply the appropriate identity and substitute exact values.

Guided example 1: Compute sin(75°) exactly

Pick a sum that uses familiar angles: 75° = 45° + 30°.

Step 1: Write the identity

sin(75°) = sin(45°+30°) = sin45° cos30° + cos45° sin30°

Step 2: Substitute exact values

• sin45° = √2/2
• cos30° = √3/2
• cos45° = √2/2
• sin30° = 1/2

sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2)

Step 3: Simplify

sin(75°) = √6/4 + √2/4 = (√6 + √2)/4

Guided example 2: Compute cos(15°) exactly

Use a difference: 15° = 45° - 30°.

Step 1: Write the identity

cos(15°) = cos(45°-30°) = cos45° cos30° + sin45° sin30°

Step 2: Substitute exact values

cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2)

Step 3: Simplify

cos(15°) = √6/4 + √2/4 = (√6 + √2)/4

Notice sin(75°) and cos(15°) match, consistent with the cofunction relationship for complementary angles.

Transforming expressions with mixed angles

When you see something like sin(x+30°) or cos(2x-45°), the goal is often to rewrite it in terms of sin x and cos x (and known constants). This is not a “products to sums” situation; it’s directly a sum/difference identity situation.

Guided example 3: Rewrite sin(x + 30°) in terms of sin x and cos x

Step 1: Apply the sum identity

sin(x+30°) = sin x cos30° + cos x sin30°

Step 2: Substitute exact constants

sin(x+30°) = sin x (√3/2) + cos x (1/2)

Step 3: Optional factoring

sin(x+30°) = (√3/2)sin x + (1/2)cos x

Guided example 4: Rewrite cos(2x - 45°)

Step 1: Apply the difference identity for cosine

cos(2x-45°) = cos(2x)cos45° + sin(2x)sin45°

Step 2: Substitute exact constants

cos(2x-45°) = cos(2x)(√2/2) + sin(2x)(√2/2)

Step 3: Factor if helpful

cos(2x-45°) = (√2/2)(cos(2x) + sin(2x))

This form is often useful when comparing expressions or setting up equations.

Choosing the correct formula: a quick decision guide

• If the expression is a single trig function of a combined angle, like sin(a±b), cos(a±b), or tan(a±b), use the corresponding sum/difference identity.
• If you are asked for an exact value and the angle can be written as a sum/difference of familiar angles (e.g., 75°, 15°, 105°), use sum/difference identities and substitute exact values.
• If the expression is already a product like sin a cos b, that is a different pattern (product-to-sum). In this chapter’s practice, you should not switch to product/sum conversions; stay with sum/difference identities when the input is a combined angle.

Practice set (focus: sum/difference selection, exact values, and rewriting)

A. Exact values

• Compute exactly: cos(75°).
• Compute exactly: sin(15°).
• Compute exactly: tan(75°) using tan(45°+30°) and exact values.
• Compute exactly: cos(105°) by writing 105°=60°+45°.

B. Rewrite in terms of sin x and cos x

• Rewrite: cos(x+60°).
• Rewrite: sin(x-45°).
• Rewrite: cos(3x+30°) (leave sin(3x), cos(3x) as-is; only expand the +30°).
• Rewrite: sin(2x-60°) (leave sin(2x), cos(2x) as-is).

C. Identity-based simplification (no product-to-sum)

• Simplify: sin(x+30°) - sin x cos30°.
• Simplify: cos(x-45°) - cos x cos45°.
• Show that: sin(x+30°) - sin(x-30°) = cos x by expanding both sides using sum/difference identities and simplifying.