Trig Identity Simplifier (basic patterns)
Recognizes the 30 most-used trigonometric identities and shows the simplified form. Pattern-library based — not a general symbolic algebra system.
Result
- Patternpythag
- Left-hand sidesin²x + cos²x
- Equivalent form1
- Identity name / noteFundamental Pythagorean identity — comes from x² + y² = 1 on the unit circle.
- Numeric check (x = 30°)LHS = 1.000000 · RHS = 1.000000 → ✓ equal
Step-by-step
- Look up canonical identity from the pattern library.
- The LHS and RHS are mathematically equal for all valid x (or A, B).
- Numeric verification at x = 30°: LHS = 1.000000 · RHS = 1.000000 → ✓ equal
How to use this calculator
- Identify the pattern in your expression (e.g. "I see sin²x + cos²x" or "I see sin(A+B)").
- Pick that pattern from the dropdown.
- Read the canonical simplified form on the right.
- For single-variable identities (Pythagorean, double-angle, etc.) the optional numeric x verifies the equality numerically.
About this calculator
A library of the 30 most-commonly-tested trigonometric identities — the ones you actually use in calculus, physics, and signal processing. NOT a general symbolic-algebra engine (which would require parsing arbitrary expressions and applying rewrite rules à la SymPy / Wolfram Alpha). Pick the pattern you see, get the canonical simplified form + a short historical / pedagogical note. The Pythagorean identity sin²x + cos²x = 1 is the foundational one — it expresses the unit-circle equation x² + y² = 1 in trigonometric form. Double-angle, sum/difference, product-to-sum, and half-angle formulas are derived from it via algebraic manipulation. For full symbolic simplification of arbitrary expressions, use a CAS (computer algebra system).