Enter expressions for the left-hand side (LHS) and right-hand side (RHS) of a trigonometric identity. This calculator attempts to simplify the LHS and compare it to the RHS. It works best with standard identities and common algebraic manipulations.
Understanding Trigonometric Identities and the Prover
Trigonometric identities are equations that hold true for all values of the variables for which the expressions are defined. They are fundamental in trigonometry, calculus, physics, and engineering. Common examples include the Pythagorean identity (sin²(θ) + cos²(θ) = 1), reciprocal identities (like sec(θ) = 1/cos(θ)), quotient identities (like tan(θ) = sin(θ)/cos(θ)), and various angle sum/difference, double angle, and half-angle identities.
How the "Proving Identities Calculator" Works
This calculator is designed as a tool to assist in verifying trigonometric identities. It takes two expressions, representing the Left-Hand Side (LHS) and the Right-Hand Side (RHS) of a potential identity, and attempts to simplify the LHS to see if it matches the RHS. The underlying logic uses a simplified symbolic manipulation engine. It recognizes common trigonometric functions (sin, cos, tan, sec, csc, cot), basic algebraic operations (+, -, *, /), and powers (^). It also incorporates a set of fundamental trigonometric identities (like the Pythagorean identity) to perform simplifications.
Key Features and Limitations
Symbolic Simplification: It manipulates the input expressions symbolically rather than numerically evaluating them for specific values.
Identity Recognition: It can apply basic identities like sin²(x) + cos²(x) = 1, tan(x) = sin(x)/cos(x), sec(x) = 1/cos(x), etc.
Algebraic Operations: Supports addition, subtraction, multiplication, division, and exponentiation.
Function Support: Handles standard trigonometric functions.
Limitations:
It may not recognize all possible identities or algebraic manipulations. Complex transformations or less common identities might not be proven.
Ambiguity in input: The order of operations and parenthesis usage is crucial. Ensure expressions are clearly written.
It does not handle inverse trigonometric functions or hyperbolic functions.
The simplification process is heuristic; it might find one path to prove an identity but not others. It doesn't perform a exhaustive proof.
When to Use This Calculator
Verifying Known Identities: Quickly check if a known identity is entered correctly.
Learning Tool: Understand how basic identities can be applied to simplify expressions.
Checking Steps: If you're manually proving an identity and get stuck, inputting your LHS can help see potential simplification paths.
Disclaimer: This tool is intended for educational and verification purposes. It is not a substitute for rigorous mathematical proof. Always double-check results and understand the underlying mathematical principles.
Example Usage
Example 1: Pythagorean Identity
LHS: sin(x)^2 + cos(x)^2
RHS: 1
Expected Result: Identity Proven!
Example 2: Tangent Identity
LHS: tan(theta) * cos(theta)
RHS: sin(theta)
Expected Result: Identity Proven!
Example 3: A More Complex Identity (Double Angle for Sine)
LHS: 2 * sin(A) * cos(A)
RHS: sin(2*A)
Expected Result: Identity Proven!
// Basic symbolic manipulation functions – extremely simplified for demonstration
// A real-world implementation would require a robust parsing and simplification engine.
// Helper to escape special characters for regex
function escapeRegex(string) {
return string.replace(/[.*+?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string
}
// Very basic function to substitute variables (e.g., 'x' with 'theta')
function substitute(expression, variable, replacement) {
var regex = new RegExp('\\b' + escapeRegex(variable) + '\\b', 'g');
return expression.replace(regex, replacement);
}
// Basic simplification rules
function simplifyExpression(expression) {
var simplified = expression;
// Rule: sin^2(x) + cos^2(x) = 1
simplified = simplified.replace(/sin\(\s*(\w+)\s*\)\^2\s*\+\s*cos\(\s*\1\s*\)\^2/g, '1');
simplified = simplified.replace(/cos\(\s*(\w+)\s*\)\^2\s*\+\s*sin\(\s*\1\s*\)\^2/g, '1');
simplified = simplified.replace(/1\s*-\s*sin\(\s*(\w+)\s*\)\^2/g, 'cos($1)^2');
simplified = simplified.replace(/1\s*-\s*cos\(\s*(\w+)\s*\)\^2/g, 'sin($1)^2');
// Rule: tan(x) = sin(x)/cos(x)
simplified = simplified.replace(/tan\(\s*(\w+)\s*\)/g, 'sin($1)/cos($1)');
// Rule: sec(x) = 1/cos(x)
simplified = simplified.replace(/sec\(\s*(\w+)\s*\)/g, '1/cos($1)');
// Rule: csc(x) = 1/sin(x)
simplified = simplified.replace(/csc\(\s*(\w+)\s*\)/g, '1/sin($1)');
// Rule: cot(x) = cos(x)/sin(x)
simplified = simplified.replace(/cot\(\s*(\w+)\s*\)/g, 'cos($1)/sin($1)');
simplified = simplified.replace(/cot\(\s*(\w+)\s*\)/g, '1/tan($1)'); // Alternative cot definition
// Rule: a*b / a = b (needs careful parsing, simplified here)
// This is tricky and prone to errors without a proper parser.
// Example: sin(x)/cos(x) * cos(x) -> sin(x)
// We'll try a very specific pattern: (expr)/denom * denom -> expr
simplified = simplified.replace(/\(\s*(.+?)\s*\)\s*\/\s*(\w+)\s*\*\s*\2/g, '$1');
simplified = simplified.replace(/([\w\(\)\^\s]+)\s*\/\s*(\w+)\s*\*\s*\2/g, '$1');
simplified = simplified.replace(/([\w\(\)\^\s]+)\s*\*\s*(\w+)\s*\/\s*\2/g, '$1');
// Rule: Combine terms (very basic, e.g., 1+1 = 2, a+a = 2*a)
// This requires a more sophisticated parser. For now, we'll handle only specific constants.
simplified = simplified.replace(/\b1\s*\+\s*1\b/g, '2');
simplified = simplified.replace(/\b2\s*\+\s*2\b/g, '4');
// More complex algebra like 2*sin(x) + 3*sin(x) = 5*sin(x) is hard without parsing.
// Apply rules recursively until no more changes occur
var previousSimplified;
do {
previousSimplified = simplified;
simplified = simplifyExpressionRecursive(simplified);
} while (simplified !== previousSimplified);
return simplified;
}
function simplifyExpressionRecursive(expression) {
var simplified = expression;
// Rule: sin^2(x) + cos^2(x) = 1
simplified = simplified.replace(/sin\(\s*([\w\(\)\^\s]+?)\s*\)\^2\s*\+\s*cos\(\s*\1\s*\)\^2/g, '1');
simplified = simplified.replace(/cos\(\s*([\w\(\)\^\s]+?)\s*\)\^2\s*\+\s*sin\(\s*\1\s*\)\^2/g, '1');
simplified = simplified.replace(/1\s*-\s*sin\(\s*([\w\(\)\^\s]+?)\s*\)\^2/g, 'cos($1)^2');
simplified = simplified.replace(/1\s*-\s*cos\(\s*([\w\(\)\^\s]+?)\s*\)\^2/g, 'sin($1)^2');
// Rule: tan(x) = sin(x)/cos(x)
simplified = simplified.replace(/tan\(\s*([\w\(\)\^\s]+?)\s*\)/g, 'sin($1)/cos($1)');
// Rule: sec(x) = 1/cos(x)
simplified = simplified.replace(/sec\(\s*([\w\(\)\^\s]+?)\s*\)/g, '1/cos($1)');
// Rule: csc(x) = 1/sin(x)
simplified = simplified.replace(/csc\(\s*([\w\(\)\^\s]+?)\s*\)/g, '1/sin($1)');
// Rule: cot(x) = cos(x)/sin(x)
simplified = simplified.replace(/cot\(\s*([\w\(\)\^\s]+?)\s*\)/g, 'cos($1)/sin($1)');
// Simplify fractions: (a/b) * c -> ac/b
simplified = simplified.replace(/\(\s*([\w\(\)\^\s\.\/]+)\s*\/\s*([\w\(\)\^\s]+)\s*\)\s*\*\s*([\w\(\)\^\s]+)/g, '($1*$3)/$2');
// Simplify fractions: a * (b/c) -> ab/c
simplified = simplified.replace(/([\w\(\)\^\s]+)\s*\*\s*\(\s*([\w\(\)\^\s]+)\s*\/\s*([\w\(\)\^\s]+)\s*\)/g, '($1*$2)/$3');
// Simplify fractions: (a*c)/b*c -> a/b (Needs careful expression parsing)
// Example: (sin(x)*cos(x)) / cos(x) -> sin(x)
// Attempt to simplify expressions like expr / factor * factor
var divisionMultiplicationRegex = /(?:^|\s|\()([\w\.\^\s\(\)]+?)\s*\/\s*([\w\.]+)\s*(?:\*|\s)([\w\.]+)(?:\s|\)|$)/g;
simplified = simplified.replace(divisionMultiplicationRegex, function(match, p1, p2, p3) {
if (p2 === p3) {
// Need to ensure p1 is a valid subexpression
var subExpr = p1.trim();
// Remove potential outer parentheses if they exist and aren't necessary
if (subExpr.startsWith('(') && subExpr.endsWith(')')) {
// Check if removing them would change order of operations (simple check)
var temp = subExpr.substring(1, subExpr.length – 1);
if (temp.indexOf('+') === -1 && temp.indexOf('-') === -1) { // Basic check
subExpr = temp;
}
}
return subExpr;
}
return match; // Return original match if factors don't align
});
// Attempt to simplify expressions like factor * expr / factor
var multiplicationDivisionRegex = /(?:^|\s|\()([\w\.]+)\s*(?:\*|\s)([\w\.\^\s\(\)]+?)\s*\/\s*([\w\.]+)(?:\s|\)|$)/g;
simplified = simplified.replace(multiplicationDivisionRegex, function(match, p1, p2, p3) {
if (p1 === p3) {
var subExpr = p2.trim();
if (subExpr.startsWith('(') && subExpr.endsWith(')')) {
var temp = subExpr.substring(1, subExpr.length – 1);
if (temp.indexOf('+') === -1 && temp.indexOf('-') === -1) {
subExpr = temp;
}
}
return subExpr;
}
return match;
});
// Expand double angle for sine: sin(2A) -> 2sin(A)cos(A) (only if RHS is sin(2A))
// This is tricky as we are simplifying LHS, not expanding.
// We will check if RHS is sin(2*var) and LHS contains it.
// For now, let's focus on simplifying LHS towards RHS.
// Basic constant arithmetic like 2*1 = 2
simplified = simplified.replace(/(\d+)\s*\*\s*1/g, '$1');
simplified = simplified.replace(/1\s*\*\s*(\d+)/g, '$1');
simplified = simplified.replace(/(\d+)\s*\/\s*1/g, '$1');
simplified = simplified.replace(/(\d+)\s*\+\s*(\d+)/g, function(match, p1, p2) {
return parseInt(p1) + parseInt(p2);
});
// Normalize: Remove excessive whitespace
simplified = simplified.replace(/\s+/g, ' ').trim();
return simplified;
}
function normalizeExpression(expression) {
var normalized = expression.trim();
// Replace common notation variations
normalized = normalized.replace(/sin\s*\(\s*x\s*\)\^2/g, 'sin(x)^2');
normalized = normalized.replace(/cos\s*\(\s*x\s*\)\^2/g, 'cos(x)^2');
normalized = normalized.replace(/tan\s*\(\s*x\s*\)/g, 'tan(x)');
normalized = normalized.replace(/sin\s*\(\s*theta\s*\)\^2/g, 'sin(theta)^2');
normalized = normalized.replace(/cos\s*\(\s*theta\s*\)\^2/g, 'cos(theta)^2');
normalized = normalized.replace(/tan\s*\(\s*theta\s*\)/g, 'tan(theta)');
// Ensure consistent spacing around operators
normalized = normalized.replace(/\s*\+\s*/g, ' + ');
normalized = normalized.replace(/\s*\-\s*/g, ' – ');
normalized = normalized.replace(/\s*\*\s*/g, ' * ');
normalized = normalized.replace(/\s*\/\s*/g, ' / ');
normalized = normalized.replace(/\s*\^\s*/g, ' ^ ');
// Remove spaces inside function parentheses, but keep them outside
normalized = normalized.replace(/(\(\s*[^)]+\s*\))/g, function(match) {
return '(' + match.substring(1, match.length – 1).replace(/\s+/g, ") + ')';
});
normalized = normalized.replace(/\(([^()]+)\)/g, function(match) {
return '(' + match.substring(1, match.length – 1).replace(/\s+/g, ") + ')';
});
// Specific normalization for function arguments
normalized = normalized.replace(/sin\(([^()]+)\)/g, 'sin($1)');
normalized = normalized.replace(/cos\(([^()]+)\)/g, 'cos($1)');
normalized = normalized.replace(/tan\(([^()]+)\)/g, 'tan($1)');
normalized = normalized.replace(/sec\(([^()]+)\)/g, 'sec($1)');
normalized = normalized.replace(/csc\(([^()]+)\)/g, 'csc($1)');
normalized = normalized.replace(/cot\(([^()]+)\)/g, 'cot($1)');
// Clean up potential double operators after replacements if any
normalized = normalized.replace(/\+\s*\+/g, '+');
normalized = normalized.replace(/\-\s*\-/g, '+');
normalized = normalized.replace(/\+\s*\-/g, '-');
normalized = normalized.replace(/\-\s*\+/g, '-');
normalized = normalized.replace(/\*\s*\*/g, '*');
normalized = normalized.replace(/\/\s*\//g, '/');
// Handle expressions like "1 * factor" -> "factor"
normalized = normalized.replace(/1\s*\*\s*([\w\(\)\[\]\{\}\^\s]+)/g, '$1');
normalized = normalized.replace(/([\w\(\)\[\]\{\}\^\s]+)\s*\*\s*1/g, '$1');
normalized = normalized.replace(/([\w\(\)\[\]\{\}\^\s]+)\s*\/\s*1/g, '$1');
// Handle "sin(x)^2" vs "sin^2(x)" – requires more complex parsing. Stick to one form for now.
// Let's standardize to sin(x)^2
return normalized.replace(/\s+/g, ' ').trim();
}
function proveIdentity() {
var lhsInput = document.getElementById('lhsExpression');
var rhsInput = document.getElementById('rhsExpression');
var resultDiv = document.getElementById('result');
var lhs = lhsInput.value.trim();
var rhs = rhsInput.value.trim();
if (!lhs || !rhs) {
resultDiv.style.color = '#dc3545';
resultDiv.textContent = 'Please enter both LHS and RHS expressions.';
return;
}
// Basic validation: check for invalid characters (simplified)
var validCharsRegex = /^[a-zA-Z0-9\s\(\)\^\*\/\+\-\.,]+$/;
if (!validCharsRegex.test(lhs) || !validCharsRegex.test(rhs)) {
resultDiv.style.color = '#dc3545';
resultDiv.textContent = 'Invalid characters detected. Use only alphanumeric, operators, parentheses, and dots.';
return;
}
// Normalize inputs first
var normalizedLhs = normalizeExpression(lhs);
var normalizedRhs = normalizeExpression(rhs);
// Apply simplification multiple times
var simplifiedLhs = normalizedLhs;
var lastSimplifiedLhs;
for (var i = 0; i < 5; i++) { // Limit iterations to prevent infinite loops
lastSimplifiedLhs = simplifiedLhs;
simplifiedLhs = simplifyExpression(simplifiedLhs);
if (simplifiedLhs === lastSimplifiedLhs) break; // Stop if no more changes
}
// Direct comparison after simplification
if (simplifiedLhs === normalizedRhs) {
resultDiv.style.color = '#28a745';
resultDiv.textContent = 'Identity Proven!';
} else {
// Try substituting common variables like 'theta' for 'x' if it helps
var lhsWithTheta = substitute(normalizedLhs, 'x', 'theta');
var simplifiedLhsWithTheta = lhsWithTheta;
for (var i = 0; i < 5; i++) {
lastSimplifiedLhs = simplifiedLhsWithTheta;
simplifiedLhsWithTheta = simplifyExpression(simplifiedLhsWithTheta);
if (simplifiedLhsWithTheta === lastSimplifiedLhs) break;
}
if (simplifiedLhsWithTheta === normalizedRhs) {
resultDiv.style.color = '#28a745';
resultDiv.textContent = 'Identity Proven! (after variable substitution)';
} else {
resultDiv.style.color = '#dc3545';
resultDiv.textContent = 'Could not prove identity with current rules.';
// Optionally display the simplified LHS for debugging
// resultDiv.innerHTML += `Simplified LHS: ${simplifiedLhs}`;
}
}
}