Prove Trig Identities Calculator

Calculator Inputs

Left Side Expression

Right Side Expression

Variable

Angle Unit

Start Angle

End Angle

Sample Count

Tolerance

Decimal Precision

Supported Expression Rules

Use explicit multiplication, such as 2*sin(x). Use ^ for powers. Supported functions include sin, cos, tan, sec, csc, cot, asin, acos, atan, sqrt, abs, ln, log, and exp.

Formula Used

The calculator compares both sides by evaluating this difference:

d(x) = |L(x) - R(x)|

If the maximum sampled difference is less than or equal to the tolerance, the identity is marked as likely true for the tested values.

Common identities used for proof planning include:

sin^2(x) + cos^2(x) = 1
1 + tan^2(x) = sec^2(x)
1 + cot^2(x) = csc^2(x)
tan(x) = sin(x) / cos(x)
sec(x) = 1 / cos(x)
csc(x) = 1 / sin(x)
cot(x) = cos(x) / sin(x)

How to Use This Calculator

Enter the left side of the identity.
Enter the right side of the identity.
Select the variable used in both expressions.
Choose radians or degrees.
Set the test range, sample count, tolerance, and precision.
Press the check button to view the result above the form.
Use the CSV or PDF button to save the report.

Example Data Table

Left Side	Right Side	Suggested Strategy	Expected Result
`sin(x)^2 + cos(x)^2`	`1`	Apply the main Pythagorean identity.	Likely identity
`tan(x)`	`sin(x)/cos(x)`	Rewrite tangent with sine and cosine.	Likely identity
`sec(x)^2 - tan(x)^2`	`1`	Rearrange `1 + tan^2(x) = sec^2(x)`.	Likely identity
`1 + cot(x)^2`	`csc(x)^2`	Use reciprocal and Pythagorean identities.	Likely identity

About Proving Trig Identities

Trigonometric identities describe equal expressions that stay equal for every allowed angle. They are essential in algebra, calculus, physics, waves, engineering, and geometry. A proof usually starts with one side and transforms it until it matches the other side. This calculator supports that habit. It checks both sides across chosen angles and reports the largest difference.

Why Identity Testing Helps

Manual proof work can fail because of one sign, reciprocal, or domain mistake. Numeric testing cannot replace a formal proof, yet it quickly finds many errors. When a proposed identity fails at even one valid sample, it cannot be universally true. When all samples pass within tolerance, the expression is likely correct over the tested range.

Useful Proof Strategy

Start with the complicated side. Rewrite tangent, cotangent, secant, and cosecant using sine and cosine. Then combine fractions, factor common terms, and apply Pythagorean identities. Avoid dividing by expressions that may become zero. Domain exclusions matter because two formulas can match numerically while still requiring different allowed angles.

Reading the Result

The calculator evaluates the left expression and right expression at evenly spaced angles. It marks skipped points where a denominator or function is undefined. It also shows the absolute difference. A small maximum difference means both sides matched within the selected tolerance. A larger value means the identity needs correction or a tighter domain.

Advanced Checking Options

The range, sample count, unit mode, tolerance, and precision fields let you match classroom or project needs. Wider ranges reveal periodic issues. More samples catch hidden failures near asymptotes. Precision controls display only, while tolerance controls the pass decision. Keep notes from the hint box beside your written proof.

Best Practice

Use exact symbols carefully. Write multiplication with an asterisk. Use powers with the caret sign. Test radians for textbook identities unless your expression was written for degrees. Increase sample count for stronger checking. Lower tolerance for precise expressions. Then use the displayed hints to write a clean step by step algebraic proof.

Final Proof Check

Before final submission, compare the calculator result with your algebraic line work. Strong identities should survive several ranges and units. Suspect identities often fail quickly when signs, squares, or reciprocal terms are misplaced.

FAQs

Can this calculator prove every trig identity?

It checks identities numerically and gives proof hints. It does not replace a formal symbolic proof. Use it to find errors and support your algebraic work.

Which angle unit should I choose?

Choose radians for most textbook identities. Choose degrees when your expression or class problem is written specifically in degrees.

Why are some sample points skipped?

Points are skipped when an expression becomes undefined. This often happens with tangent, cotangent, secant, cosecant, or division by zero.

What tolerance should I use?

A small tolerance, such as 0.000001, works well for most checks. Use a larger value when expressions create rounding noise.

How do I enter powers?

Use the caret symbol. For example, write sine squared as sin(x)^2. Always include parentheses around function inputs.

Can I use theta instead of x?

Yes. Select theta as the variable. The calculator also accepts x, theta, and t as simple variable names.

Why did a true identity fail?

Check missing multiplication symbols, wrong units, domain problems, or tolerance settings. Also confirm both sides use the same variable.

What should I write in my proof?

Use the hints to choose identities. Then rewrite one side step by step until it matches the other side.