A trig identity is an equation that holds for every angle. There are dozens in textbooks, but almost all of them come from a handful. Learn these well and derive the rest when you need them. The unit circle guide shows where the values come from.
Reciprocal and quotient identities
| Identity | |
|---|---|
| csc θ = 1/sin θ | sec θ = 1/cos θ |
| cot θ = 1/tan θ | tan θ = sin θ/cos θ |
| cot θ = cos θ/sin θ |
These are definitions, so they’re the first thing to reach for when an expression mixes functions: rewrite everything in sine and cosine.
The Pythagorean identity
sin²θ + cos²θ = 1
On the unit circle, a point at angle θ is (cos θ, sin θ), and its distance from the origin is 1. The Pythagorean theorem gives the identity directly.
Divide every term by cos²θ or by sin²θ to get the other two:
- 1 + tan²θ = sec²θ
- cot²θ + 1 = csc²θ
Example sin θ = 3/5 and θ is in quadrant II. Find cos θ and tan θ.
- cos²θ = 1 − 9/25 = 16/25, so cos θ = ±4/5.
- Cosine is negative in quadrant II: cos θ = −4/5.
- tan θ = sin θ/cos θ = (3/5)/(−4/5) = −3/4.
Ignoring the quadrant
The square root gives ±. The quadrant decides the sign: all positive in I, only sine in II, only tangent in III, only cosine in IV.
Even and odd identities
- cos(−θ) = cos θ
- sin(−θ) = −sin θ
- tan(−θ) = −tan θ
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
Note the sign flip in the cosine version: cos(A + B) has a minus.
Example Find the exact value of sin 75°
- 75° = 45° + 30°.
- sin 75° = sin 45° cos 30° + cos 45° sin 30°.
- = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
Example Find the exact value of cos 15°
- 15° = 45° − 30°.
- cos 15° = cos 45° cos 30° + sin 45° sin 30° = (√6 + √2)/4.
That makes sense: cos 15° = sin 75°, because the two angles add to 90°.
Double-angle identities
Set A = B in the sum identities:
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
Example sin θ = 3/5 in quadrant II, so cos θ = −4/5. Find sin 2θ and cos 2θ.
- sin 2θ = 2(3/5)(−4/5) = −24/25.
- cos 2θ = 1 − 2(9/25) = 7/25.
sin 2θ is not 2 sin θ
sin 60° ≈ 0.866, but 2 sin 30° = 1. The 2 inside the function changes the angle, not the output.
How to prove an identity
- Work on one side only, usually the more complicated one.
- Rewrite everything in sine and cosine.
- Combine fractions, factor, or use sin²θ + cos²θ = 1.
- Stop when that side matches the other.
You can’t move terms across the equals sign, because you don’t know yet that the two sides are equal. That’s what you’re proving.
Example Prove tan θ + cot θ = sec θ csc θ
- Left side: sin θ/cos θ + cos θ/sin θ.
- Common denominator: (sin²θ + cos²θ)/(sin θ cos θ).
- The top is 1: 1/(sin θ cos θ).
- = (1/cos θ)(1/sin θ) = sec θ csc θ.
Example Prove (1 − cos²θ)/sin θ = sin θ
- 1 − cos²θ = sin²θ.
- sin²θ/sin θ = sin θ.
Check yourself
cos θ = 5/13 and θ is in quadrant IV. Find sin θ.
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−12/13.
tan θ = 3/4 and θ is in quadrant I. Find sec θ.
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5/4. 1 + 9/16 = 25/16.
Find the exact value of cos 75°.
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(√6 − √2)/4.
sin θ = 4/5, cos θ = 3/5. Find sin 2θ.
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24/25.
Simplify sin θ · cot θ.
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cos θ.
Simplify (sec²θ − 1)/tan θ.
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tan θ. sec²θ − 1 = tan²θ.