Trigonometric sum and difference identities are powerful tools for simplifying complex expressions. They allow us to break down trig functions of combined angles into simpler components, making calculations easier and revealing hidden patterns in periodic functions.

These identities form the foundation for more advanced trig concepts. By mastering them, we gain the ability to manipulate and analyze trigonometric expressions with greater ease, opening doors to applications in physics, engineering, and beyond.

Sum and Difference Identities

Cosine sum and difference applications

• Cosine sum formula $\cos(A + B) = \cos A \cos B - \sin A \sin B$ expresses the cosine of the sum of two angles and simplifies expressions involving the cosine of a sum (45° + 60°)
• Cosine difference formula $\cos(A - B) = \cos A \cos B + \sin A \sin B$ expresses the cosine of the difference of two angles and simplifies expressions involving the cosine of a difference (90° - 30°)
• Applying the formulas involves substituting the given angles for A and B in the appropriate formula and simplifying the resulting expression using basic trigonometric values or identities (Pythagorean identity)
• These formulas are particularly useful when working with angles expressed in radian measure

Sine sum and difference usage

• Sine sum formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$ represents the sine of the sum of two angles and simplifies expressions involving the sine of a sum (30° + 45°)
• Sine difference formula $\sin(A - B) = \sin A \cos B - \cos A \sin B$ represents the sine of the difference of two angles and simplifies expressions involving the sine of a difference (60° - 15°)
• Applying the formulas involves substituting the given angles for A and B in the appropriate formula and simplifying the resulting expression using basic trigonometric values or identities (special right triangle ratios)
• These formulas are essential when analyzing periodic functions

Tangent sum and difference simplification

• Tangent sum formula $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ expresses the tangent of the sum of two angles and simplifies expressions involving the tangent of a sum (0° + 45°)
• Tangent difference formula $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ expresses the tangent of the difference of two angles and simplifies expressions involving the tangent of a difference (90° - 60°)
• Applying the formulas involves substituting the given angles for A and B in the appropriate formula and simplifying the resulting expression using basic trigonometric values or identities (reciprocal identities)

Cofunction Identities and Proof Techniques

Cofunction identities with sum-difference formulas

• Cofunction identities relate trigonometric functions of complementary angles:
  • $\sin(90^\circ - \theta) = \cos \theta$ (30°)
  • $\cos(90^\circ - \theta) = \sin \theta$ (45°)
  • $\tan(90^\circ - \theta) = \cot \theta$ (60°)
  • $\cot(90^\circ - \theta) = \tan \theta$ (75°)
  • $\sec(90^\circ - \theta) = \csc \theta$ (15°)
  • $\csc(90^\circ - \theta) = \sec \theta$ (0°)
• Combining cofunction identities with sum and difference formulas involves substituting complementary angles into sum and difference formulas and simplifying the resulting expression using cofunction identities ($\cos(60^\circ + (90^\circ - \theta))$)
• These identities are often visualized using the unit circle

Sum-difference formulas in identity proofs

• Verifying identities starts with the more complex side of the identity, applies sum and difference formulas to simplify the expression, uses other trigonometric identities and algebraic techniques as needed, and aims to transform the complex side into the simpler side of the identity ($\sin(A+B)\cos(A-B)$)
• Proof techniques include:
  1. Direct proof: Start with the left side and manipulate it to match the right side
  2. Indirect proof: Assume the identity is false and derive a contradiction
  3. Showing equivalence: Manipulate both sides independently to arrive at the same expression

Additional Angle Formulas

• Double angle formulas express trigonometric functions of 2θ in terms of functions of θ
• Half angle formulas express trigonometric functions of θ/2 in terms of functions of θ
• These formulas are derived from the sum and difference identities and are useful for simplifying complex trigonometric expressions