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MonoCalc

Trigonometric Identity Calculator

Calculation Mode

Input

Use: sin, cos, tan, cot, sec, csc. Use ² or ^2 for powers.

Example Expressions

💡 Quick Tips

Powered by mathjs

• Use sin²(x) or sin^2(x) or sin(x)^2 for powers

• Supported functions: sin, cos, tan, cot, sec, csc

• Advanced expressions: 2*sin(x)*cos(x), (sin(x) + cos(x))^2

• Click any identity in the reference sheet to load it

• Simplify mode uses symbolic algebra for accurate results

• Verify mode tests both symbolic and numerical equality

• Evaluate mode supports complex nested expressions

• Half-angle, product-to-sum, and power reduction identities supported

About the tool

📐 Trigonometric Identity Calculator – Simplify, Verify & Evaluate Trig Identities

The Trigonometric Identity Calculator is a powerful tool for working with trigonometric identities, powered by mathjs for advanced symbolic algebra and precise simplification. Whether you need to simplify complex trigonometric expressions, verify that two expressions are equal, or evaluate identities at specific angles, this calculator provides instant results with detailed step-by-step explanations.

This comprehensive guide explains what trigonometric identities are, how they work, and provides practical applications for mathematics, physics, engineering, and advanced calculus.

📘 What are Trigonometric Identities?

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variables (where both sides are defined). Unlike trigonometric equations that may have specific solutions, identities are universal truths in trigonometry.

For example, the famous Pythagorean identity sin²(x) + cos²(x) = 1 holds true for any angle x. This means if you calculate sin²(30°) + cos²(30°), sin²(45°) + cos²(45°), or any other angle, the result will always be exactly 1.

🎯 Main Categories of Trigonometric Identities

1. Pythagorean Identities

These fundamental identities are derived from the Pythagorean theorem applied to the unit circle:

  • sin²(x) + cos²(x) = 1 – The most fundamental identity
  • 1 + tan²(x) = sec²(x) – Derived by dividing the first identity by cos²(x)
  • 1 + cot²(x) = csc²(x) – Derived by dividing the first identity by sin²(x)

2. Reciprocal Identities

These define the relationships between the six trigonometric functions:

  • csc(x) = 1/sin(x) – Cosecant is the reciprocal of sine
  • sec(x) = 1/cos(x) – Secant is the reciprocal of cosine
  • cot(x) = 1/tan(x) – Cotangent is the reciprocal of tangent

3. Quotient Identities

These express tangent and cotangent in terms of sine and cosine:

  • tan(x) = sin(x)/cos(x)
  • cot(x) = cos(x)/sin(x)

4. Double Angle Formulas

These identities relate trigonometric functions of an angle to functions of twice that angle:

  • sin(2x) = 2sin(x)cos(x)
  • cos(2x) = cos²(x) - sin²(x) (or equivalently: 2cos²(x) - 1, or 1 - 2sin²(x))
  • tan(2x) = 2tan(x)/(1 - tan²(x))

5. Sum and Difference Formulas

These express functions of sums and differences of angles:

  • 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))

⚙️ How the Trigonometric Identity Calculator Works

This calculator offers three powerful modes, leveraging mathjs for advanced symbolic manipulation:

1. Simplify Mode (Enhanced with Symbolic Algebra)

Enter any trigonometric expression and the calculator will automatically simplify it using symbolic algebra and known identities. For example:

  • Input: sin²(x) + cos²(x) → Output: 1
  • Input: 1 + tan²(x) → Output: sec²(x)
  • Input: sin(x)/cos(x) → Output: tan(x)
  • Input: 2*sin(x)*cos(x) → Output: sin(2*x) (double angle formula)
  • Input: (sin(x) + cos(x))^2 → Output: 1 + 2*sin(x)*cos(x) (expansion)

The calculator shows each simplification step, explaining which identity was applied at each stage. The mathjs engine applies over 18 different identity rules automatically.

2. Verify Identity Mode (Symbolic + Numerical Testing)

Enter two expressions to verify if they are identically equal. The calculator will:

  1. Simplify both sides independently using symbolic algebra
  2. Compare the simplified forms
  3. If forms differ, test numerical equality at multiple sample points

This dual approach is perfect for:

  • Homework problems asking you to prove identities
  • Checking your work on identity transformations
  • Discovering relationships between different trig expressions
  • Handling complex expressions that simplify differently

3. Evaluate Mode (Full Expression Support)

Calculate the numeric value of any trigonometric expression at a specific angle. You can enter angles in either degrees or radians. The mathjs parser handles complex nested expressions like:

  • (1 - cos(2*x))/2 at x = 45°
  • sin(x)^4 + cos(x)^4 at x = π/4
  • tan(x) + cot(x) at x = 30°

This mode is useful for:

  • Verifying identities numerically at specific angles
  • Checking if your simplified expression gives correct values
  • Understanding how identities work with concrete numbers

📊 Practical Applications

Mathematics & Calculus

Trigonometric identities are essential for integration, solving equations, and simplifying complex expressions. For example, the double-angle formula cos(2x) = 1 - 2sin²(x) can be rearranged to sin²(x) = (1 - cos(2x))/2, which is crucial for integrating sin²(x).

Physics & Engineering

Identities appear throughout wave mechanics, alternating current circuits, signal processing, and mechanics. The sum formulas are used extensively in analyzing superposition of waves and interference patterns.

Navigation & Surveying

Sum and difference formulas help calculate distances and bearings when dealing with multiple angles, essential for GPS systems and land surveying.

Computer Graphics

Rotation matrices and transformations rely heavily on trigonometric identities, particularly for 3D graphics, game development, and computer animation.

🔍 Using the Calculator Effectively

Input Format

You can write expressions in several ways:

  • Powers: Use sin²(x) or sin^2(x) or sin(x)^2
  • Functions: sin, cos, tan, cot, sec, csc
  • Operators: +, -, *, /
  • Parentheses: Use freely to group terms: (sin(x) + 1)²

Step-by-Step Solutions

Each calculation shows detailed steps explaining the transformation process. This helps you:

  • Learn which identities to apply in different situations
  • Understand the logical progression of simplifications
  • Verify your manual work against automated solutions
  • Prepare for exams by studying the methodology

Identity Reference Sheet

The built-in reference sheet contains all major trigonometric identities organized by category. Click any identity to instantly load it into the calculator, perfect for quick reference or learning.

💡 Tips & Tricks

  • Always start simplification by looking for Pythagorean identities – they're the most powerful
  • When verifying identities, try working with the more complex side first
  • Convert all functions to sine and cosine when stuck – it often reveals simplification paths
  • Use Evaluate mode to verify your simplifications numerically at specific angles
  • Remember that tan(x) = sin(x)/cos(x) is often the key to unlocking complex expressions
  • For exam preparation, practice both simplifying and verifying identities

🚀 Advanced Features (Powered by mathjs)

This calculator supports advanced mathematical operations:

  • Symbolic Simplification: Automatically applies over 18 trigonometric identity rules including Pythagorean, quotient, reciprocal, double angle, half-angle, and power reduction formulas
  • Nested Expressions: Handle complex expressions like sin(cos(x)), (sin(x) + cos(x))^2
  • Higher Powers: Simplify sin³(x), cos⁴(x), and other power expressions
  • Product Expansions: Automatically expand 2*sin(x)*cos(x) to sin(2*x)
  • Dual Verification: Test identities both symbolically and numerically for maximum confidence
  • Complex Evaluations: Evaluate any valid mathematical expression at any angle
  • Step-by-step Explanations: See exactly which rules were applied and when
  • Both degrees and radians for numerical evaluation
  • Copy results to clipboard for easy sharing
  • Expandable step-by-step solutions

🎓 Common Mistakes to Avoid

  • Don't confuse sin²(x) with sin(x²) – they are completely different!
  • Remember that (sin(x))² = sin²(x), not sin(2x)
  • When dividing by trigonometric functions, watch for undefined values (e.g., tan(90°) is undefined)
  • Always check domain restrictions when simplifying or verifying
  • Don't forget the ± signs in identities derived from square roots

🔗 Related Concepts

Understanding trigonometric identities is essential for success in advanced mathematics. They connect to:

  • Trigonometric equations – Use identities to solve for angles
  • Calculus – Integration techniques, derivatives, and limits
  • Complex numbers – Euler's formula relates trig functions to exponentials
  • Fourier analysis – Decomposing signals into trigonometric components
  • Differential equations – Trig functions as solutions

📚 Example Problems

Example 1: Simplify sin²(x) + cos²(x)

Solution: This is the fundamental Pythagorean identity. Result: 1

Example 2: Verify that 1 + tan²(x) = sec²(x)

Steps:

  • Start with sin²(x) + cos²(x) = 1
  • Divide both sides by cos²(x)
  • Get: sin²(x)/cos²(x) + 1 = 1/cos²(x)
  • Simplify: tan²(x) + 1 = sec²(x) ✓

Example 3: Simplify sin(x)/cos(x)

Solution: Using the quotient identity, this simplifies to tan(x)

Example 4: Evaluate sin²(30°) + cos²(30°)

Steps:

  • sin(30°) = 0.5, so sin²(30°) = 0.25
  • cos(30°) ≈ 0.866, so cos²(30°) ≈ 0.75
  • Sum: 0.25 + 0.75 = 1.0 ✓

🚀 Advanced Features

This calculator supports:

  • Nested expressions: sin(cos(x))
  • Complex powers: sin³(x), cos⁴(x)
  • Multiple function combinations: sin(x)cos(x) + tan(x)
  • Both degrees and radians for numerical evaluation
  • Copy results to clipboard for easy sharing
  • Expandable step-by-step solutions

🎯 Summary

The Trigonometric Identity Calculator is an essential tool for students, educators, engineers, and anyone working with trigonometry. Whether you're learning identities for the first time, solving complex calculus problems, or verifying your manual calculations, this calculator provides accurate results with clear explanations every step of the way.

Frequently Asked Questions

  • Is the Trigonometric Identity Calculator free ?

    Yes, Trigonometric Identity Calculator is totally free :)

  • Can i use the Trigonometric Identity Calculator offline ?

    Yes, you can install the webapp as PWA.

  • Is it safe to use Trigonometric Identity Calculator ?

    Yes, any data related to Trigonometric Identity Calculator only stored in your browser(if storage required). You can simply clear browser cache to clear all the stored data. We do not store any data on server.

  • What is a trigonometric identity?

    A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variables. Common examples include sin²(x) + cos²(x) = 1 (Pythagorean identity), 1 + tan²(x) = sec²(x), and sin(2x) = 2sin(x)cos(x). These identities are fundamental tools for simplifying expressions and solving trigonometric equations.

  • What are the main types of trigonometric identities?

    The main types include: (1) Pythagorean identities like sin²(x) + cos²(x) = 1; (2) Reciprocal identities like csc(x) = 1/sin(x); (3) Quotient identities like tan(x) = sin(x)/cos(x); (4) Co-function identities; (5) Even-odd identities; (6) Sum and difference formulas; (7) Double angle formulas like sin(2x) = 2sin(x)cos(x); and (8) Half-angle formulas.

  • How do I verify if two trigonometric expressions are equal?

    To verify if two expressions are identical, you can: (1) Simplify both sides independently using known identities; (2) Transform one side into the other using algebraic manipulation and trig identities; (3) Evaluate both expressions for multiple angle values to check numerical equality; or (4) Use this calculator to automatically simplify and compare expressions symbolically.

  • What is the difference between simplifying and verifying identities?

    Simplifying means reducing a trigonometric expression to its most basic form (e.g., simplifying sin²(x) + cos²(x) to 1). Verifying means proving that two expressions are equal by transforming one into the other or showing they simplify to the same result. This calculator can do both: simplify a single expression or verify that two expressions are identical.

  • Can I evaluate trigonometric identities for specific angles?

    Yes! While identities are true for all angles, you can evaluate them at specific values to verify numerically. For example, if you evaluate sin²(30°) + cos²(30°), you'll get 0.25 + 0.75 = 1, confirming the Pythagorean identity. This calculator supports evaluation mode where you can input an angle in degrees or radians to compute numeric results.

  • What are the Pythagorean trigonometric identities?

    The three Pythagorean identities are: (1) sin²(x) + cos²(x) = 1 (fundamental), (2) 1 + tan²(x) = sec²(x), and (3) 1 + cot²(x) = csc²(x). These are derived from the Pythagorean theorem and are among the most frequently used identities for simplification and problem-solving in trigonometry.