Math formulas, level 3: identities and rules
15 rules from algebra, trigonometry and logarithms. Exponent rules assume the base is not zero where dividing. Typed answers can use ^ for powers.
Identities are statements that are true for every value of the letters. They are the tools for simplifying and solving in algebra, trigonometry and beyond, and they show up on every high school and college entrance test.
Each rule has a reason you can check with small numbers: try a = 3 and b = 2 in (a + b)² and see why the middle term 2ab must be there. Checking with numbers is retrieval practice and proof at once. Chunk by family (special products, exponents, trigonometry, logarithms), and notice that the log rules are the exponent rules in disguise.
Encode make it vivid → Space bring it back on day 1, 3, 7, 21 → Retrieve pull it from your head. From the infographic Remember It on Purpose.
Math formulas, level 3: identities and rules
| Formula for | Formula | Letters stand for |
|---|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² | any numbers a and b |
| Square of a difference | (a − b)² = a² − 2ab + b² | any numbers a and b |
| Difference of squares | (a − b)(a + b) = a² − b² | any numbers a and b |
| Product of powers | aᵐ · aⁿ = aᵐ⁺ⁿ | a = base, m and n = exponents |
| Quotient of powers | aᵐ / aⁿ = aᵐ⁻ⁿ | a = base (not 0), m and n = exponents |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | a = base, m and n = exponents |
| Zero exponent | a⁰ = 1 | a = any base except 0 |
| Negative exponent | a⁻ⁿ = 1 / aⁿ | a = base (not 0), n = exponent |
| Pythagorean identity | sin²θ + cos²θ = 1 | θ = any angle |
| Tangent as a quotient | tan θ = sin θ / cos θ | θ = any angle where cos θ ≠ 0 |
| Double-angle formula for sine | sin 2θ = 2 sin θ cos θ | θ = any angle |
| Double-angle formula for cosine | cos 2θ = cos²θ − sin²θ | θ = any angle (also 1 − 2sin²θ or 2cos²θ − 1) |
| Product rule for logarithms | log_b(MN) = log_b(M) + log_b(N) | b = base, M and N = positive numbers |
| Quotient rule for logarithms | log_b(M/N) = log_b(M) − log_b(N) | b = base, M and N = positive numbers |
| Power rule for logarithms | log_b(Mⁿ) = n · log_b(M) | b = base, M = a positive number, n = any number |
Each rule has a reason you can check with small numbers: try a = 3 and b = 2 in (a + b)² and see why the middle term 2ab must be there. Checking with numbers is retrieval practice and proof at once. Chunk by family (special products, exponents, trigonometry, logarithms), and notice that the log rules are the exponent rules in disguise. · teddyrookbook.com/memory/challenges/math-identities/
Sources
- OpenStax Intermediate Algebra 2e, Chapter 5 Key Concepts
- OpenStax Algebra and Trigonometry, Chapter 9 Key Equations
- OpenStax College Algebra, Chapter 6 Key Equations
Checked October 30, 2026. Progress is kept on this device only (per learner). More lists: all memory challenges · memory games.