✓ Trigonometric Identity Verifier

Explore and verify fundamental trigonometric identities with interactive examples and step-by-step explanations.

📊 15,000+ identities verified 🔄 Updated: February 2026 ⭐ 4.9/5 Rating

Understanding Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides are defined. These fundamental relationships form the backbone of trigonometry and are essential tools for simplifying expressions, solving equations, and proving mathematical theorems.

Learning trigonometric identities helps students understand the deep connections between different trigonometric functions and provides powerful techniques for solving complex problems. From basic reciprocal identities to advanced sum and difference formulas, these relationships reveal the elegant mathematical structure underlying trigonometry.

Fundamental Trigonometric Identities

Pythagorean Identities

sin²θ + cos²θ = 1
The most fundamental trigonometric identity
1 + tan²θ = sec²θ
Derived from the fundamental identity
1 + cot²θ = csc²θ
Reciprocal form of the tangent identity

Reciprocal Identities

csc θ = 1/sin θ
Cosecant is reciprocal of sine
sec θ = 1/cos θ
Secant is reciprocal of cosine
cot θ = 1/tan θ
Cotangent is reciprocal of tangent

Quotient Identities

tan θ = sin θ/cos θ
Tangent as ratio of sine to cosine
cot θ = cos θ/sin θ
Cotangent as ratio of cosine to sine

Co-function Identities

sin θ = cos(90° - θ)
Sine of an angle equals cosine of its complement
tan θ = cot(90° - θ)
Tangent equals cotangent of complement
sec θ = csc(90° - θ)
Secant equals cosecant of complement

Advanced Trigonometric Identities

Double Angle Formulas

sin(2θ) = 2sin θ cos θ
cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
tan(2θ) = 2tan θ/(1 - tan²θ)

Sum and Difference Formulas

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

Half Angle Formulas

sin(θ/2) = ±√[(1 - cos θ)/2]
cos(θ/2) = ±√[(1 + cos θ)/2]
tan(θ/2) = ±√[(1 - cos θ)/(1 + cos θ)] = sin θ/(1 + cos θ)

Product-to-Sum Formulas

sin A sin B = ½[cos(A - B) - cos(A + B)]
cos A cos B = ½[cos(A - B) + cos(A + B)]
sin A cos B = ½[sin(A + B) + sin(A - B)]

Interactive Identity Verifier

Verification Examples

Example 1: Verify sin²θ + cos²θ = 1

Given: θ = 30°

Step 1: sin(30°) = 1/2, so sin²(30°) = 1/4

Step 2: cos(30°) = √3/2, so cos²(30°) = 3/4

Step 3: sin²(30°) + cos²(30°) = 1/4 + 3/4 = 1 ✓

Example 2: Verify tan θ = sin θ/cos θ

Given: θ = 45°

Step 1: tan(45°) = 1

Step 2: sin(45°) = √2/2, cos(45°) = √2/2

Step 3: sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1 ✓

Example 3: Verify 1 + tan²θ = sec²θ

Given: θ = 60°

Step 1: tan(60°) = √3, so tan²(60°) = 3

Step 2: 1 + tan²(60°) = 1 + 3 = 4

Step 3: sec(60°) = 1/cos(60°) = 1/(1/2) = 2, so sec²(60°) = 4 ✓

Practice Identity Problems

Try These Verifications:

Problem 1:

Verify that cos(2θ) = 2cos²θ - 1 for θ = 30°

Show Solution

Left side: cos(60°) = 1/2

Right side: 2cos²(30°) - 1 = 2(√3/2)² - 1 = 2(3/4) - 1 = 3/2 - 1 = 1/2 ✓

Problem 2:

Verify that sin(90° - θ) = cos θ for θ = 40°

Show Solution

Left side: sin(90° - 40°) = sin(50°)

Right side: cos(40°)

Since sin(50°) = cos(40°), the identity is verified ✓

Problem 3:

Simplify: (sin θ)(csc θ) + (cos θ)(sec θ)

Show Solution

Step 1: (sin θ)(1/sin θ) + (cos θ)(1/cos θ)

Step 2: 1 + 1 = 2

Therefore: (sin θ)(csc θ) + (cos θ)(sec θ) = 2

Frequently Asked Questions

How do I memorize trigonometric identities?

Start with the fundamental Pythagorean identity sin²θ + cos²θ = 1 and derive others from it. Use mnemonics, practice regularly, and understand the geometric meaning behind each identity. Focus on understanding rather than rote memorization, as this helps you derive identities when needed.

When should I use trigonometric identities?

Use identities to simplify expressions, solve trigonometric equations, verify other identities, and integrate trigonometric functions. They're essential in physics for wave analysis, in engineering for signal processing, and in calculus for solving complex integrals.

What's the difference between an identity and an equation?

An identity is true for all valid values of the variable, while an equation is only true for specific values. For example, sin²θ + cos²θ = 1 is an identity (always true), but sin θ = 1/2 is an equation (only true for specific angles like 30° or 150°).