Worked example

Verify or refute: sin²θ · sec²θ − 1 = tan²θ

Steps

1. sin²θ · sec²θ − 1
   Start with the left side only — the standard advice.
2. = sin²θ · (1/cos²θ) − 1
   Reciprocal identity: sec θ = 1/cos θ. · Reciprocal identity (secant)
3. = sin²θ/cos²θ − 1
   Multiply.
4. = tan²θ − 1
   Quotient identity: tan θ = sin θ/cos θ. · Quotient identity (tangent)
5. compare with the right side, tan²θ
   They differ by 1, so this is NOT an identity — the verifier would show a counterexample at any angle, e.g. θ = 0.6 gives 0.36 vs 0.45.

Worked example used on the verifier page: the point is that a failed verification is as useful as a passing one, and a counterexample proves it.

Identities used

• Reciprocal identity (secant) — Convert sec θ to 1/cos θ whenever it shares an expression with tan θ, since tan θ is already sin/cos and a common denominator then appears.
• Quotient identity (tangent) — The single most common first step in a verification: replace tan θ with sin θ/cos θ so the whole expression is in one function family and fractions can be combined.

Check it yourself

Open the verifier with the main identity pre-filled — substituting your own numbers into a step is the fastest way to find out which line you did not follow.

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