Trig Identities Practice: Interactive Quiz, Worksheet & Reference Sheet
Pick a level and a mode, then answer ten questions. Practice explains each answer as you go, Quiz scores you at the end, and Match pairs expressions with their simplified forms. Your best score is saved on your own device.
A trig identity is an equation that is true for every angle where both sides are defined. Learning them is less about memorising a long list and more about recognising which one turns a messy expression into a simple one. The drill below gives you ten random questions at a time, with a worked explanation after every answer.
Everything here covers Algebra 2 and Precalculus: the reciprocal, quotient and Pythagorean identities first, then cofunction, sum and difference, double angle, power reducing and half angle. The printable worksheet and reference sheet are free, with the full answer key included.
Free Printable Trig Identities PDFs
Both printables are designed for black-and-white printing — nothing depends on colour. Choose US Letter or A4.
Trig Identities Worksheet with Answer Key
30 questions over four pages: simplify to a single function, factor and combine, double and half angle, then four identities to verify with ruled working space. The full answer key includes a worked route through every proof.
Trig Identities Reference Sheet
Every identity for Algebra 2 and Precalculus on a single page, grouped by family and labelled by course level, with a short note on the four steps for verifying an identity.
The Trig Identities You Need
Algebra 2 identities
| Family | Identities |
|---|---|
| Reciprocal | csc θ = 1 / sin θ · sec θ = 1 / cos θ · cot θ = 1 / tan θ |
| Quotient | tan θ = sin θ / cos θ · cot θ = cos θ / sin θ |
| Pythagorean | sin²θ + cos²θ = 1 · 1 + tan²θ = sec²θ · 1 + cot²θ = csc²θ |
| Even–odd | sin(−θ) = −sin θ · cos(−θ) = cos θ · tan(−θ) = −tan θ |
Only the first Pythagorean identity is worth memorising. Divide it by cos²θ to get 1 + tan²θ = sec²θ, or by sin²θ to get 1 + cot²θ = csc²θ.
Precalculus identities
| Family | Identities |
|---|---|
| Cofunction | sin(π/2 − θ) = cos θ · cos(π/2 − θ) = sin θ · tan(π/2 − θ) = cot θ |
| Sum & difference | 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 | sin 2θ = 2 sin θ cos θ cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1 tan 2θ = 2 tan θ/(1 − tan²θ) |
| Power reducing | sin²θ = (1 − cos 2θ)/2 · cos²θ = (1 + cos 2θ)/2 |
| Half angle | sin(θ/2) = ±√[(1 − cos θ)/2] cos(θ/2) = ±√[(1 + cos θ)/2] tan(θ/2) = (1 − cos θ)/sin θ |
The ± on a half-angle formula is decided by the quadrant that θ/2 lands in, not the quadrant of θ. The tangent version needs no ± at all.
How to Verify a Trig Identity in Four Steps
- Work on one side only — the messier one. An identity is not an equation to solve. You transform one side until it looks exactly like the other, and nothing ever crosses the equals sign.
- Rewrite everything in sin and cos. When you are stuck, this almost always unsticks you. Secant, cosecant, tangent and cotangent all come apart into sine and cosine.
- Look for a Pythagorean identity, a common denominator, or a difference of squares. Those three account for most of the work in a typical problem.
- Check the domain. An identity only holds where both sides are defined, so note any angle that makes a denominator zero.
Worked Examples
Eight examples, easiest first. Open one to see the full working.
1. Simplify sin θ · csc θ
csc θ is the reciprocal of sin θ, so sin θ · csc θ = sin θ · (1/sin θ) = 1, for every θ where sin θ ≠ 0.
2. Simplify tan θ · cos θ
Rewrite the tangent: tan θ · cos θ = (sin θ / cos θ) · cos θ. The cosines cancel, leaving sin θ.
3. Simplify (1 + sin θ)(1 − sin θ)
This is a difference of squares: 1 − sin²θ. By the Pythagorean identity that equals cos²θ.
4. Simplify sin⁴θ − cos⁴θ
Factor as a difference of squares: (sin²θ − cos²θ)(sin²θ + cos²θ). The second bracket is 1, so the answer is sin²θ − cos²θ — which is also −cos 2θ.
5. Verify (sec θ − cos θ) / tan θ = sin θ
Start with the left side. sec θ − cos θ = 1/cos θ − cos θ = (1 − cos²θ)/cos θ = sin²θ/cos θ.
Now divide by tan θ = sin θ/cos θ, which means multiplying by cos θ/sin θ: (sin²θ/cos θ) · (cos θ/sin θ) = sin θ. ✓
6. Verify csc θ sec θ − cot θ = tan θ
Left side in sine and cosine: 1/(sin θ cos θ) − cos θ/sin θ. Put it over the common denominator sin θ cos θ: (1 − cos²θ)/(sin θ cos θ).
The numerator is sin²θ, so this is sin²θ/(sin θ cos θ) = sin θ/cos θ = tan θ. ✓
7. Verify cot²θ − cos²θ = cot²θ · cos²θ
Factor cos²θ out of the left side: cos²θ(1/sin²θ − 1) = cos²θ · (1 − sin²θ)/sin²θ.
Since 1 − sin²θ = cos²θ, that is cos²θ · cos²θ/sin²θ = cot²θ cos²θ. ✓
8. Verify sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2 csc θ
Common denominator sin θ(1 + cos θ) gives [sin²θ + (1 + cos θ)²] / [sin θ(1 + cos θ)].
Expand the numerator: sin²θ + 1 + 2cos θ + cos²θ = 2 + 2cos θ = 2(1 + cos θ). Cancel the bracket: 2/sin θ = 2 csc θ. ✓
Six Common Mistakes
- Treating it like an equation. You do not solve an identity. You transform one side until it matches the other.
- Moving a term across the equals sign. Everything you do stays on the side you chose to work on.
- Cancelling across a sum. In (1 + cos θ)/cos θ the cosines do not cancel — only a factor of the whole numerator can.
- Forgetting the ± on a half angle. The sign comes from the quadrant of θ/2, not of θ.
- Using the wrong form of cos 2θ. All three forms are correct; pick the one that matches what is already on the page.
- Stopping early. Keep going until both sides are written in exactly the same form, not merely a similar one.
Trig Identities FAQ
What are the basic 8 trig identities?
The basic eight are the three reciprocal identities (csc, sec, cot), the two quotient identities (tan and cot in terms of sine and cosine), and the three Pythagorean identities. Nearly every Algebra 2 simplification problem uses only these.
Which trig identities do I actually have to memorise?
The reciprocal and quotient identities, sin²θ + cos²θ = 1, and the sum and difference formulas. The other two Pythagorean identities come from dividing the first one, the double angle formulas come from the sum formulas with A = B, and the power reducing and half angle formulas come from the double angle ones.
What is the difference between verifying and solving?
Verifying means showing an equation holds for every angle in the domain, so you rewrite one side until it matches the other. Solving means finding the specific angles that make an equation true, and there you may move terms across the equals sign.
Which side should I start from?
The more complicated one. There is more to simplify, and you have a clear target to aim at. If you are stuck after a few lines, rewrite everything in sine and cosine and try again.
Do trig identities hold for every angle?
Only where both sides are defined. tan θ = sin θ/cos θ fails at θ = π/2, for instance, because cos θ = 0 there. A complete answer notes those excluded values.
Can the practice tool mark a proof for me?
No — a proof can be written many valid ways, so no automatic marker can judge one fairly. The drill covers simplifying, completing an identity and matching; the proof work lives in the printable worksheet, which includes a worked route through each one.
Related Practice and Tools
- Unit Circle Game — drill the exact values the identities keep asking you for.
- Unit Circle Guide — the full chart, quadrant signs and reference angles.
- Unit Circle Practice — timed recall of sine, cosine and tangent values.
- Right Triangle Calculator — where the ratios behind the identities come from.
- Law of Sines Worksheet — apply sine to triangles with no right angle, including the ambiguous case.
- Math Worksheets — every printable on the site, with worked answer keys.