Mathematics July 20, 2026 · 12 min read

Trigonometric Identities Explained: Pythagorean, Double-Angle, and More

Explore all major trigonometric identities including Pythagorean, double-angle, sum/difference, and reciprocal identities. Verify them with our calculator.

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. They form the backbone of trigonometry, enabling you to simplify complex expressions, solve equations, and prove mathematical theorems. From the fundamental Pythagorean identity to the powerful double-angle formulas, mastering these identities is essential for success in mathematics, physics, and engineering.

Key Takeaway

The Pythagorean identity sin²θ + cos²θ = 1 is the most fundamental trig identity. From it, you can derive all other Pythagorean identities and many compound formulas. Identities are proven through algebraic manipulation, not by testing specific angles.

1. Pythagorean Identities

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

The first identity is derived directly from the Pythagorean theorem applied to the unit circle. The other two follow by dividing the first identity by cos²θ and sin²θ respectively.

2. Double-Angle Identities

sin(2θ) = 2sinθ·cosθ
cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan(2θ) = 2tanθ / (1 - tan²θ)

Double-angle formulas are special cases of the sum formulas where A = B. They are derived by setting A = B in the sum identities and simplifying. The cosine double-angle has three equivalent forms, each useful in different situations.

3. Reciprocal and Quotient Identities

  • Reciprocal: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
  • Quotient: tan θ = sin θ/cos θ, cot θ = cos θ/sin θ
  • Cofunction: sin θ = cos(90° - θ), tan θ = cot(90° - θ)

4. How to Verify a Trig Identity

  1. Start with one side of the equation (usually the more complex side).
  2. Convert everything to sine and cosine.
  3. Apply Pythagorean identities to simplify.
  4. Factor, cancel, or combine fractions as needed.
  5. Continue until both sides match.

5. Frequently Asked Questions

Are trig identities always true?

Yes, trigonometric identities are equations that hold for all values in their domain. This distinguishes them from trigonometric equations, which are only true for specific angle values.

How many trig identities are there?

There are infinitely many, but mastering the core categories (Pythagorean, Reciprocal, Quotient, Sum/Difference, Double-Angle, Half-Angle) allows you to derive many others.

Try Our Calculator

Use our free Trig Identities Calculator to verify Pythagorean, double-angle, and reciprocal identities.