Reciprocal and quotient identities · trigonometric identity
tan θ = sin θ / cos θ — Quotient identity (tangent)
When to use it
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.
Why it is true
In the unit circle, tan θ is the slope of the radius to the point (cos θ, sin θ), and a slope is rise over run: sin θ/cos θ.
The full line-by-line version is on the proof page for quotient identity (tangent); the “how would I find this myself” version is in the derivation.
Where it comes from
Cancel the hypotenuse. If you need to remove tangent from an expression, the question to ask is 'what is tangent made of?' — two ratios whose denominator cancels.
- tan θ = opp/adjDefinition.
- opp/adj = (opp/hyp)/(adj/hyp)Multiply top and bottom by 1/hyp.
- = sin θ/cos θRecognise both ratios.
Worked examples
- Verify or refute: sin²θ · sec²θ − 1 = tan²θsin²θ · sec²θ − 1 = tan²θ, an identity (true for every θ where defined).
- Verify tan θ + cot θ = sec θ csc θConvert to sin and cos, take the common denominator, and the Pythagorean identity closes it.
- Verify tan A + tan B = sin(A + B)/(cos A cos B)Common denominator, then recognise the sine sum.
Checked at these angles
Two sides of the identity evaluated with the same engine that runs the verifier. A single line disagreeing would break the build.
| Angle | Left side | Right side | Agree |
|---|---|---|---|
| 0° | 0 | 0 | yes |
| 7.5° | 0.131652 | 0.131652 | yes |
| 15° | 0.267949 | 0.267949 | yes |
| 18° | 0.32492 | 0.32492 | yes |
| 22.5° | 0.414214 | 0.414214 | yes |
| 30° | 0.57735 | 0.57735 | yes |
| 37° | 0.753554 | 0.753554 | yes |
| 45° | 1 | 1 | yes |
| 53° | 1.327045 | 1.327045 | yes |
| 60° | 1.732051 | 1.732051 | yes |
| 67.5° | 2.414214 | 2.414214 | yes |
| 75° | 3.732051 | 3.732051 | yes |
| 120° | -1.732051 | -1.732051 | yes |
| 135° | -1 | -1 | yes |
| 150° | -0.57735 | -0.57735 | yes |
| 180° | -0 | -0 | yes |
| 210° | 0.57735 | 0.57735 | yes |
| 240° | 1.732051 | 1.732051 | yes |
| 300° | -1.732051 | -1.732051 | yes |
| 330° | -0.57735 | -0.57735 | yes |
| 360° | -0 | -0 | yes |
The mistake students make
Applying it where cos θ = 0. tan 90° has no value, and the right side has the same problem — 1/0 — which is why the identity is a definition and not a coincidence.
Try it
- Verify this identity — both sides are pre-filled; change one character and watch the counterexample appear.
- Ask “which identity should I use?” about the left-hand side — this identity is what the chooser should rank first.
- Cheat sheet entry for reciprocal and quotient identities.
Related identities
- cot θ = cos θ / sin θQuotient identity (cotangent)detailsproof
- sec θ = 1/cos θReciprocal identity (secant)detailsproof
- 1 + tan²θ = sec²θPythagorean identity (tangent form)detailsproof
Category hub: Reciprocal and quotient identities · all identities: /identities