Fundamental Identities Review

Before tackling advanced identities, ensure you know these foundations:

sin^2θ + cos^2θ = 1

Pythagorean identity

tanθ = sinθ / cosθ

Quotient identity

1 + tan^2θ = sec^2θ

Derived from Pythagorean

1 + cot^2θ = csc^2θ

Derived from Pythagorean

Compound Angle Formulas

The compound angle (or addition) formulas express trigonometric functions of a sum or difference in terms of the individual angles:

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

Memory aid: For sine, the signs match the operation (sin(A+B) has a +). For cosine, the signs are opposite (cos(A+B) has a -).

Proving Trigonometric Identities

To prove an identity, you must show that the left-hand side (LHS) equals the right-hand side (RHS). Key strategies include:

Work on one side only (usually the more complex side) and transform it into the other.
Convert everything to sin and cos if the expression involves tan, sec, csc, or cot.
Use known identities such as Pythagorean, compound angle, and double angle formulas.
Factor or expand as needed; look for common factors or difference of squares.
Combine fractions using a common denominator when the expression has multiple terms.

Important: Never move terms across the equals sign. You must work on one side and transform it independently until it matches the other side.

Key Vocabulary

Compound Angle

An expression involving the sum or difference of two angles, such as sin(A + B).

Identity

An equation that is true for all valid values of the variable, not just specific ones.

Double Angle

A special case of compound angles where both angles are the same: sin(2A) = sin(A + A).

Reciprocal Functions

secθ = 1/cosθ, cscθ = 1/sinθ, cotθ = 1/tanθ.

Worked Examples

1

Find the exact value of sin(75°).

Step 1: Write 75° = 45° + 30°

Step 2: sin(75°) = sin(45° + 30°) = sin45°cos30° + cos45°sin30°

Step 3: = (√2/2)(√3/2) + (√2/2)(1/2)

Answer: sin(75°) = (√6 + √2) / 4

2

Prove that cos(A - B) - cos(A + B) = 2 sinA sinB.

LHS: cos(A - B) - cos(A + B)

= [cosA cosB + sinA sinB] - [cosA cosB - sinA sinB]

= cosA cosB + sinA sinB - cosA cosB + sinA sinB

= 2 sinA sinB

= RHS (identity proven)

3

Simplify sin(x + π/6) + sin(x - π/6).

Step 1: Expand both: [sinx cos(π/6) + cosx sin(π/6)] + [sinx cos(π/6) - cosx sin(π/6)]

Step 2: The cosx sin(π/6) terms cancel.

Step 3: = 2 sinx cos(π/6) = 2 sinx (√3/2)

Answer: √3 sinx

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

Using the compound angle formula, what is cos(A + B)?

cosA cosB - sinA sinB cosA cosB + sinA sinB sinA cosB + cosA sinB cosA sinB - sinA cosB

Question 2

What is the exact value of cos(15°)?

(√6 - √2) / 4 (√6 + √2) / 4 √3/2 (√2 - √6) / 4

Question 3

When proving an identity, you should:

Cross-multiply both sides Substitute a specific angle to check Work on one side and transform it to match the other Move all terms to one side and simplify to zero

Question 4

sin(A + B) + sin(A - B) simplifies to:

2 sinA sinB 2 cosA cosB 2 cosA sinB 2 sinA cosB

Question 5

Which identity is equivalent to sin^2θ?

1 - cos^2θ 1 + cos^2θ cos^2θ - 1 tan^2θ / sec^2θ

Advanced Trigonometric Identities