Sum-to-Product Formulas
Trigonometric Identities for Simplifying Expressions
Sum/Difference to Product
These formulas convert sums or differences of trigonometric functions into products,
making them easier to simplify, solve, or integrate. They’re particularly useful in calculus
and trigonometric equation solving.
Cosine Sum
\[ \cos x + \cos y = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2} \]
Use: Converts sum of cosines to product of cosines.
Cosine Difference
\[ \cos x – \cos y = -2 \sin \frac{x+y}{2} \sin \frac{x-y}{2} \]
Note: Negative sign appears in front.
Sine Sum
\[ \sin x + \sin y = 2 \sin \frac{x+y}{2} \cos \frac{x-y}{2} \]
Pattern: Sine of average × cosine of half-difference.
Sine Difference
\[ \sin x – \sin y = 2 \cos \frac{x+y}{2} \sin \frac{x-y}{2} \]
Pattern: Cosine of average × sine of half-difference.
Memory Tip
Cosine formulas: Both start with cosine on the left ⇒ both have cosine in first product term.
Sine formulas: Both start with sine on the left ⇒ both have sine in first product term.
The sign in cosine difference is negative; all others are positive.
Trigonometry Reference • Sum-to-Product Formulas
Useful for integration, simplification, and equation solving