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Trigonometric Identity Calculator

Verify trig identities (like sin²x + cos²x = 1) or simplify trig expressions with student-friendly rewrite steps plus a numeric sanity check. Supports π/pi, √/sqrt(), powers (like sin^2(x)), and common trig functions (sin, cos, tan, sec, csc, cot).

Background

A trig identity is an equation that’s true for all x in its domain. This calculator can verify identities by testing many x-values (skipping undefined points), and can simplify expressions by applying common identity rewrites toward a target form.

Enter an identity or expression

Tip: For “prove that …”, use Verify. For “simplify …”, use Simplify.

Verify identity

Generates explainable rewrite steps (not a full CAS proof), then checks numerically.

Use: sin(), cos(), tan(), sec(), csc(), cot(), pi/π, sqrt(). Example: tan(x) = sin(x)/cos(x)

Options

Most trig identities use radians by default.

Affects numeric tolerance for “true”.

Chips prefill and calculate immediately.

Identity library

Click a rule to apply to the focused input (or LHS)

Pythagorean

Reciprocal / Quotient

Double-angle (advanced)

Result

No results yet. Enter an identity/expression and click Calculate.

How to use this calculator

  • Choose Verify to test LHS = RHS, or Simplify to rewrite a single expression.
  • Use Quick picks to load a common identity instantly.
  • Use the Identity library to apply one rule to the focused box (or LHS by default), then click Calculate.
  • If values blow up or become undefined, check the Domain warnings section.

How this calculator works

  • Step engine: applies common identity rewrites based on your selected goal/target.
  • Numeric check: evaluates both forms at multiple “smart” x-values (skipping undefined points).
  • Not a full CAS: steps are explainable “breadcrumbs,” and the numeric check is a strong sanity check.

Formulas & Identities Used

Pythagorean: sin²x + cos²x = 1

Quotient: tan x = sin x / cos x, cot x = cos x / sin x

Reciprocal: sec x = 1 / cos x, csc x = 1 / sin x

Double-angle: sin(2x)=2sin x cos x, cos(2x)=cos²x−sin²x

Example Problem & Step-by-Step Solution

Example 1 — Verify an identity

Verify sin²x + cos²x = 1.

  1. Enter LHS: sin(x)^2 + cos(x)^2, RHS: 1.
  2. Choose goal “sin & cos only”.
  3. Click Calculate → the step engine rewrites and the numeric check confirms the difference stays ~0.

Example 2 — Simplify an expression

Simplify (1 − cos²x)/sin x.

  1. Switch to Simplify mode.
  2. Enter: (1 - cos(x)^2)/sin(x).
  3. Click Calculate → uses 1 − cos²x = sin²x to get sin x.

Frequently Asked Questions

Q: Is this a full symbolic proof?

No. It generates explainable rewrite steps and then verifies numerically across many test points (skipping undefined points).

Q: Why do some x values get skipped?

Because expressions like tan x or sec x are undefined when their denominator is zero (e.g., cos x = 0).

Q: What input syntax is supported?

Use sin(x), cos(x), tan(x), sec(x), csc(x), cot(x), plus pi/π, sqrt(), and powers like sin(x)^2.