Trig Identity Calculator

Using the Trig Identity Solver

Verifying trig identities with this calculator follows a clear, repeatable process. Enter the left-hand side and right-hand side of your suspected identity into the two input fields. Select a test angle (common choices are 30°, 45°, or 60° because their exact values are well known) and click Verify. The calculator evaluates both expressions at your chosen angle and at ten additional angles spanning the full range of the unit circle, then reports whether the values match within the numerical tolerance you set.

For identities involving two variables — such as sum-to-product identities like sin(x) + sin(y) = 2 sin((x+y)/2) cos((x-y)/2) — enter a second angle under Advanced Options. The calculator substitutes both values and tests equality. When working with angle measurements, our angle calculator helps convert between degrees and radians for precise calculations.

1. Enter the left-side expression (e.g., sin²(x) + cos²(x)).
2. Enter the right-side expression (e.g., 1).
3. Choose a test angle and unit (degrees or radians).
4. Click Verify — the tool tests your primary angle plus 10 additional angles automatically.
5. Review the step-by-step results showing left-side value, right-side value, and pass/fail for each angle.

Verify Pythagorean Identity Calculator

The Pythagorean identities are derived directly from the Pythagorean theorem applied to the unit circle and form the basis for most trigonometric simplifications. Every trigonometry student should memorize all three forms:

• 1. sin²(x) + cos²(x) = 1 — the fundamental Pythagorean identity. Because any point on the unit circle has coordinates (cos x, sin x) and lies at distance 1 from the origin, the sum of squares always equals 1.
• 2. 1 + tan²(x) = sec²(x) — divide the fundamental identity by cos²(x). Valid whenever cos(x) is not zero.
• 3. 1 + cot²(x) = csc²(x) — divide the fundamental identity by sin²(x). Valid whenever sin(x) is not zero.

Use the Pythagorean identity calculator feature above to verify any of these by selecting them from the identity reference library and clicking Verify. The multi-angle test confirms the identity holds across the entire domain.

Proving Identities Calculator With Steps

While this calculator verifies identities numerically (which is not the same as a formal algebraic proof), it is an essential companion to the proving process. Before you invest time writing a formal proof, use this tool to confirm the identity actually holds. Here are the standard strategies for proving trig identities:

• Work one side only. Transform the more complex side into the simpler side using known identities. Never cross the equals sign.
• Convert everything to sine and cosine. Replace tan, cot, sec, and csc with their sin/cos definitions to find common structure.
• Factor and simplify. Look for common factors, difference of squares (a² - b²), or perfect square trinomials.
• Multiply by a conjugate. If you see (1 - sin x) in a denominator, multiply top and bottom by (1 + sin x) to create a Pythagorean identity.
• Combine fractions. Get a common denominator to combine separate fractions into one expression.

Worked Example: Prove that sec(x) - cos(x) = sin(x) tan(x)

Step 1: Verify numerically first. Enter sec(x) - cos(x) as the left side and sin(x) * tan(x) as the right side. Click Verify. The calculator confirms they match at all test angles.

Step 2: Now prove algebraically. Start with the left side:
sec(x) - cos(x) = 1/cos(x) - cos(x) = (1 - cos²(x))/cos(x) = sin²(x)/cos(x) = sin(x) · sin(x)/cos(x) = sin(x) · tan(x).

The numerical verification gave us confidence to proceed. The algebraic proof confirms the identity rigorously. For double-angle specific calculations, our double angle calculator provides specialized sin(2x), cos(2x), and tan(2x) computations.

Double Angle Identities

The double angle formulas express trigonometric functions of 2x in terms of functions of x. These are essential for integration, solving equations, and simplifying expressions:

• sin(2x) = 2 sin(x) cos(x)
• cos(2x) = cos²(x) - sin²(x) = 2cos²(x) - 1 = 1 - 2sin²(x)
• tan(2x) = 2tan(x) / (1 - tan²(x))

Sum and Difference Identities

Sum and difference identities allow you to find exact values of trig functions at non-standard angles. For example, sin(75°) = sin(45° + 30°) can be computed using the sine sum formula:

• sin(A + B) = sin A cos B + cos A sin B
• sin(A - B) = sin A cos B - cos A sin B
• cos(A + B) = cos A cos B - sin A sin B
• cos(A - B) = cos A cos B + sin A sin B

Sum-to-Product and Product-to-Sum Identities

These identities convert between sums and products of trigonometric functions. They are especially useful in solving trigonometric equations and in Fourier analysis:

• sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2)
• sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2)
• cos A + cos B = 2 cos((A+B)/2) cos((A-B)/2)
• cos A - cos B = -2 sin((A+B)/2) sin((A-B)/2)

Negative Angle and Cofunction Identities

Negative angle identities tell you the parity of each function: sin(-x) = -sin(x) (odd), cos(-x) = cos(x) (even), tan(-x) = -tan(x) (odd). Cofunction identities relate complementary angle pairs: sin(90° - x) = cos(x), tan(90° - x) = cot(x), and sec(90° - x) = csc(x). Use the calculator above to verify any of these instantly.

Students Verifying Homework and Exam Prep

Precalculus and calculus students use this trig identities calculator to check their manual work before submitting assignments. When you are asked to “verify the identity” or “establish the identity” on a worksheet, running the expressions through this tool first saves time and catches sign errors or domain mistakes that are easy to miss.

Educators Creating Problem Sets

Teachers and professors use the calculator to validate that textbook identities are stated correctly and to test new problems before assigning them. A quick numerical check prevents the frustration of students trying to prove an identity that contains a typo.

Engineers and Scientists

Electrical engineers working with AC circuits, physicists analyzing wave interference, and signal processing professionals all rely on trigonometric identities daily. This identity solver provides fast verification when simplifying complex expressions in real-world applications. When applying identities to integration problems, our trig substitution calculator uses these identities to solve complex integrals.

Establish the Identity Calculator Tips

When your textbook says “establish the identity,” it means the same thing as “prove the identity” — show that the equation holds for all values in the domain. Here are the most common mistakes students make and how to avoid them:

• Mistake: Working both sides simultaneously. In a proof, you should transform one side into the other. Working both sides and meeting in the middle is not a valid proof technique in most courses.
• Mistake: Ignoring domain restrictions. The identity 1 + tan²(x) = sec²(x) is not valid at x = 90° because both tan and sec are undefined there. Always state the domain.
• Mistake: Mixing degrees and radians. Entering 45 when the calculator is set to radians gives a completely different result than 45°. Always check your angle unit setting.
• Mistake: Assuming numerical equality proves the identity. Testing at one angle is necessary but not sufficient. Our calculator tests 10+ angles to increase confidence, but a formal algebraic proof is still required for homework.

What is the difference between verifying and proving an identity?

Verifying means testing that both sides produce the same value at specific angles — this is what our calculator does. Proving means showing algebraically that the identity holds for all values in the domain. Verification builds confidence; proof provides certainty. Use the calculator to verify first, then write your formal proof.

Can this calculator prove trigonometric identities?

This is a numerical verification tool, not a symbolic algebra system. It tests whether two expressions are equal at multiple angles. If they match at all test points, the identity is very likely correct. However, for a formal proof you still need to manipulate one side algebraically into the other using known identities.

What identities are included in the reference library?

The calculator includes 25+ identities across seven categories: Pythagorean identities, quotient and reciprocal identities, double angle formulas, sum and difference formulas, sum-to-product identities, negative angle identities, and cofunction identities. Click any identity to load it directly into the calculator for instant verification.