Overview

These formulas are useful when products appear in AIME-style problems or when you want to collapse a sum into a single trig function.

Core Skills

Decide Direction

If you see a product, use product-to-sum; if you see a sum of two sines or cosines, use sum-to-product.

Keep the $1/2$ Factor

All product-to-sum formulas include a factor of $1/2$. Include it early to avoid mistakes.

Use Special Angles

Pick angles like $15^\circ$ and $75^\circ$ so the sum or difference is a standard angle.

Product-to-Sum

$\sin\alpha\cos\beta = \tfrac12[\sin(\alpha+\beta)+\sin(\alpha-\beta)]$

$\cos\alpha\cos\beta = \tfrac12[\cos(\alpha+\beta)+\cos(\alpha-\beta)]$

$\sin\alpha\sin\beta = \tfrac12[\cos(\alpha-\beta)-\cos(\alpha+\beta)]$

Sum-to-Product

$\sin A + \sin B = 2\sin\tfrac{A+B}{2}\cos\tfrac{A-B}{2}$

$\cos A + \cos B = 2\cos\tfrac{A+B}{2}\cos\tfrac{A-B}{2}$

Worked Example

Simplify $\sin 75^\circ + \sin 15^\circ$.

$2\sin 45^\circ\cos 30^\circ = \frac{\sqrt{6}}{2}.$

More Examples

Example 1: Product to Sum

Simplify $\sin 20^\circ\cos 40^\circ$.

$\tfrac12[\sin 60^\circ+\sin(-20^\circ)] = \tfrac12(\sqrt{3}/2-\sin 20^\circ)$.

Example 2: Sum to Product

Simplify $\cos 80^\circ + \cos 40^\circ$.

$2\cos 60^\circ\cos 20^\circ = \cos 20^\circ$.

Example 3: Difference of Sines

Simplify $\sin 50^\circ-\sin 10^\circ$.

$2\cos 30^\circ\sin 20^\circ = \sqrt{3}\sin 20^\circ$.

Strategy Checklist

• Identify whether a product or sum is present.
• Apply the correct formula and keep the $1/2$.
• Simplify with special angles where possible.

Common Pitfalls

• Dropping the $1/2$ factor in product-to-sum.
• Mixing up the $A+B$ and $A-B$ halves.
• Switching sine and cosine formulas.

Practice Problems