Prove Trig Identities Calculator

Trigonometric identities are equations that hold true for all values of the variables where both sides are defined. From algebraic simplifications to calculus and physics applications, being able to prove an identity—rather than just numerically checking it—is a core skill. Yet, manual proofs can be time-consuming: picking the right identity, choosing which side to manipulate, and avoiding algebraic traps all add friction.

Prove Trig Identities Calculator

What Does “Proving an Identity” Mean?

To prove a trigonometric identity, you must show that the left-hand side (LHS) and right-hand side (RHS) are algebraically equivalent for all permissible variable values. You may transform only one side—or both sides separately—using valid trig relationships and algebraic rules until both sides match or reduce to the same simplest form.

Key idea: an identity is not solved at a particular angle; it is justified for all angles in the common domain.

Core Trigonometric Identities Used by the Calculator

Below are the most common identities (written in plain text) that the calculator leverages when constructing a proof:

Pythagorean identities

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

Reciprocal identities

  • sec(x) = 1 / cos(x)
  • csc(x) = 1 / sin(x)
  • cot(x) = 1 / tan(x) = cos(x) / sin(x)

Quotient identities

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

Even–odd identities

  • sin(-x) = -sin(x)
  • cos(-x) = cos(x)
  • tan(-x) = -tan(x)
  • csc(-x) = -csc(x)
  • sec(-x) = sec(x)
  • cot(-x) = -cot(x)

Cofunction identities (in radians)

  • sin(π/2 – x) = cos(x)
  • cos(π/2 – x) = sin(x)
  • tan(π/2 – x) = cot(x)
  • csc(π/2 – x) = sec(x)
  • sec(π/2 – x) = csc(x)
  • cot(π/2 – x) = tan(x)

Sum and difference identities

  • 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))

Double-angle identities

  • sin(2x) = 2 sin(x) cos(x)
  • cos(2x) = cos^2(x) – sin^2(x) = 1 – 2 sin^2(x) = 2 cos^2(x) – 1
  • tan(2x) = (2 tan(x)) / (1 – tan^2(x))

Half-angle identities (principal branches assumed)

  • sin^2(x) = (1 – cos(2x)) / 2
  • cos^2(x) = (1 + cos(2x)) / 2
  • tan^2(x) = (1 – cos(2x)) / (1 + cos(2x))

Product-to-sum and sum-to-product (when needed)

  • sin(a)cos(b) = (1/2)[sin(a + b) + sin(a – b)]
  • cos(a)cos(b) = (1/2)[cos(a + b) + cos(a – b)]
  • sin(a)sin(b) = (1/2)[cos(a – b) – cos(a + b)]

The calculator also applies algebraic rules: factoring, common denominators, rationalizing, and canceling common factors (only when legal).

How to Use the Prove Trig Identities Calculator

  1. Enter the identity
    Type your expression in the form LHS = RHS. Example: sec(x) - cos(x) = tan(x)sin(x).
  2. Choose a strategy (optional)
    Select options like “reduce to sine and cosine,” “work from the more complex side,” or “simplify both sides to a common target.”
  3. Select angle mode (radians or degrees)
    Identities are the same regardless, but some inputs (like cofunction angles) may be easier in radians.
  4. Click Prove
    The tool attempts a chain of transformations and shows each step with reasons (e.g., “use tan(x) = sin(x)/cos(x)” or “apply 1 + tan^2(x) = sec^2(x)”).
  5. Review steps
    Inspect the step-by-step derivation, domain notes (for example, where denominators require cos(x) ≠ 0), and the final verification.
  6. Export or copy
    Copy the proof steps to your notes or download a clean, classroom-ready sequence.

Worked Examples

Example 1: Prove 1 + tan^2(x) = sec^2(x)

Starting side: LHS

  1. tan^2(x) = (sin^2(x)) / (cos^2(x))
  2. 1 + tan^2(x) = 1 + sin^2(x)/cos^2(x)
  3. Write 1 as cos^2(x)/cos^2(x):
    1 + sin^2(x)/cos^2(x) = [cos^2(x) + sin^2(x)] / cos^2(x)
  4. Use sin^2(x) + cos^2(x) = 1:
    = 1 / cos^2(x) = sec^2(x)
    Conclusion: LHS = RHS, identity proven (for cos(x) ≠ 0).

Example 2: Prove sec(x) – cos(x) = tan(x) sin(x)

Approach: Reduce to sine and cosine.
LHS = sec(x) – cos(x)
= 1/cos(x) – cos(x)
= [1 – cos^2(x)] / cos(x)
Use 1 – cos^2(x) = sin^2(x):
= sin^2(x) / cos(x)
= sin(x) * [sin(x)/cos(x)]
= sin(x) * tan(x)
Conclusion: LHS = RHS (for cos(x) ≠ 0).

Example 3: Prove (1 – cos(2x)) / (1 + cos(2x)) = tan^2(x)

Use double-angle conversions.
Start with LHS:
(1 – cos(2x)) / (1 + cos(2x))
Replace with half-angle forms:
1 – cos(2x) = 2 sin^2(x)
1 + cos(2x) = 2 cos^2(x)
Thus LHS = [2 sin^2(x)] / [2 cos^2(x)] = tan^2(x)
Conclusion: Proven (for cos(x) ≠ 0).

Example 4: Prove sin(x) + sin(3x) = 2 sin(2x) cos(x)

Use sum formulas or product-to-sum.
sin(x) + sin(3x) = 2 sin((x + 3x)/2) cos((x – 3x)/2)
= 2 sin(2x) cos(-x)
cos(-x) = cos(x):
= 2 sin(2x) cos(x)
Conclusion: Identity holds for all x.

Tips for Successful Trig Proofs

  • Work from the “messier” side. The more complex side typically has more structure to simplify.
  • Convert to sine and cosine. Many identities reduce quickly when everything is in sin and cos.
  • Avoid illegal cancellations. Only cancel a factor if it’s not zero on the domain you’re considering.
  • Use Pythagorean identities early. Replace sin^2 and cos^2 to eliminate squares or create factors.
  • Rationalize clever denominators. Conjugates can unlock hidden Pythagorean patterns.
  • Check domains. Note exclusions like cos(x) ≠ 0 or sin(x) ≠ 0 when dividing.
  • Aim for a common target. Sometimes simplifying both sides to the same expression (like sin(x)/cos(x)) is cleaner than making one side match the other directly.

Who Is This Calculator For?

  • Students practicing identity proofs in trigonometry courses.
  • Teachers generating step-by-step classroom examples.
  • Engineers and scientists simplifying expressions before integration/differentiation.
  • Anyone who wants instant verification of proposed trig equalities.

20 Frequently Asked Questions (FAQs)

Q1. What does the Prove Trig Identities Calculator do?
It verifies trigonometric identities by transforming one or both sides using valid trig and algebraic rules until both sides match.

Q2. Does it show steps?
Yes, it lists each transformation and references the identity or algebra rule used.

Q3. Can it handle multiple variables or parameters?
Yes, as long as the expressions are symbolic and the identities are trigonometric in nature.

Q4. What angle units should I use?
Identities are unit-agnostic; however, enter angles consistently (radians or degrees) when numeric angles appear.

Q5. Will it prove identities with composite angles like sin(3x) or sin(a ± b)?
Yes, it uses sum/difference and multiple-angle formulas.

Q6. Can I input products like sin(x)cos(x) or rational forms like (1 – cos(2x))/(1 + cos(2x))?
Absolutely. The tool handles products, quotients, sums, and powers.

Q7. What if an identity is false?
The calculator will fail to reconcile both sides and may show a contradiction or a condition where the equality doesn’t hold.

Q8. Does the tool check domain restrictions?
Yes, it flags steps that require conditions like cos(x) ≠ 0 or sin(x) ≠ 0.

Q9. Can I choose which side to manipulate?
Yes. You can select “transform LHS,” “transform RHS,” or “simplify both to a common form.”

Q10. Will it always pick the shortest proof?
Not always, but it aims for a clear, logically correct sequence.

Q11. Can it simplify to only sine and cosine?
Yes, there’s an option to convert everything to sin and cos.

Q12. Does it support cofunction and even–odd identities?
Yes, it employs cofunction and parity identities when useful.

Q13. What about product-to-sum or sum-to-product conversions?
Supported. These can be crucial for non-obvious simplifications.

Q14. Can it handle absolute values or piecewise definitions?
Basic identity proofs avoid absolute value; if present, the tool will note additional conditions.

Q15. Can I export the proof?
Yes, you can copy the steps or export a formatted proof for assignments or slides.

Q16. Does it support inverse trig functions?
It can manipulate expressions with arcsin, arccos, arctan under standard identities, but some proofs may require domain notes.

Q17. Can I check equivalent forms like 1 – 2 sin^2(x) = cos(2x)?
Yes, double-angle alternatives are part of the rule set.

Q18. How exact are the transformations?
They are symbolic and exact; the tool doesn’t rely on numerical approximations to justify equality.

Q19. Is there a way to practice with random problems?
You can generate suggested identities of varying difficulty to practice proofs.

Q20. Is the calculator free to use?
Yes, it’s free and designed to help you master trigonometric proofs efficiently.

Final Thoughts

Proving trig identities is a fundamental skill that builds algebraic fluency and prepares you for advanced mathematics. The Prove Trig Identities Calculator provides transparent, step-by-step transformations that validate or refute proposed equalities while teaching the strategies behind them. Use it to check homework, design lessons, or quickly simplify expressions for problem-solving. With practice—and with clear proof steps—you’ll recognize patterns faster, avoid common pitfalls, and approach every identity with confidence.