How to Memorize Trig Identities (Only the Ones You Need)

6 min read · FlashDeck

The honest answer to how to memorize trig identities: don't memorize all of them. Lock in three core identities (sin²θ + cos²θ = 1 and the sine and cosine sum formulas) plus the basic definitions, and you can derive nearly every other identity on your formula sheet in under a minute. Then drill the handful you use constantly with flashcards so they're instant on the test.

If you keep mixing up which double-angle version has the minus sign, FlashDeck turns your identity list into cards and brings back the exact ones you keep fumbling: Build my study plan.

This works for Precalc, AP Precalc, AP Calc, IB, A-level Maths (UK) and Grade 12 math (Canada). Quick check: if the unit circle values aren't solid yet, start with how to memorize the unit circle, since identities build on it.

Which trig identities to memorize

Here's the tier list. Memorize tier 1 cold. Know how to derive tier 2. Recognize tier 3.

Check your course's formula sheet. Some exams give you a lot of these; A-level students, for instance, get a formula booklet with some identities in it. Anything provided doesn't need memorizing, just recognizing. What you do need is to spot when to use each one.

The definitions (tier 1)

• tan θ = sin θ / cos θ
• cot θ = cos θ / sin θ
• csc θ = 1 / sin θ (cosec in the UK)
• sec θ = 1 / cos θ
• cot θ = 1 / tan θ

Trick for reciprocals: each "co-" function pairs with a non-"co-" one. csc goes with sin, sec goes with cos. Weird but true, and that mismatch is exactly why people mix them up.

Derive the rest from 3 core ones

Your three core identities:

1. sin²θ + cos²θ = 1
2. sin(A ± B) = sin A cos B ± cos A sin B
3. cos(A ± B) = cos A cos B ∓ sin A sin B

Memory hooks for the sum formulas:

• Sine mixes it up and keeps the sign: sin-cos, cos-sin, same sign as the input.
• Cosine keeps them together and flips the sign: cos-cos, sin-sin, opposite sign.

Say those out loud a few times. They carry you.

Pythagorean identity tricks

From sin²θ + cos²θ = 1:

• Divide everything by cos²θ: tan²θ + 1 = sec²θ
• Divide everything by sin²θ: 1 + cot²θ = csc²θ

That's it. Two of your three Pythagorean identities come from one division each. Takes ten seconds to derive in an exam margin.

Rearranged versions you'll use constantly

• sin²θ = 1 − cos²θ
• cos²θ = 1 − sin²θ
• sec²θ − 1 = tan²θ

These show up in verifying identities and in calc integrals, so they're worth carding too.

Double-angle and sum formulas

Double-angle from the sum formulas

Set A = B = θ:

• sin(2θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ
• cos(2θ) = cos θ cos θ − sin θ sin θ = cos²θ − sin²θ

Then swap using the Pythagorean identity to get the other two versions of cos(2θ):

• cos(2θ) = 2cos²θ − 1 (replace sin²θ with 1 − cos²θ)
• cos(2θ) = 1 − 2sin²θ (replace cos²θ with 1 − sin²θ)

Power-reducing (a.k.a. why you'll love the cos 2θ versions)

Rearrange those last two:

• cos²θ = (1 + cos 2θ) / 2
• sin²θ = (1 − cos 2θ) / 2

Hook: cosine gets the plus. These come back hard in calc when you integrate sin² and cos² (more in how to memorize integration formulas).

tan(A ± B)

Divide sin(A + B) by cos(A + B), then divide top and bottom by cos A cos B:

tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Hook: top keeps the sign, bottom flips it (same energy as cosine).

Drill them with cards

Deriving is your backup. On a timed test you want the common ones instant. That's what flashcards are for.

Solid identity cards:

• Q: cos(2θ) in terms of sin only? A: 1 − 2sin²θ
• Q: What do you get dividing sin² + cos² = 1 by cos²? A: tan²θ + 1 = sec²θ
• Q: sin(A − B)? A: sin A cos B − cos A sin B
• Q: sec θ equals? A: 1 / cos θ
• Q: sin²θ written without a square? A: (1 − cos 2θ) / 2

Make one card per identity, plus "which identity would you use for..." cards. Recall is the goal; if you can see the answer before flipping, you know it.

The hard part is being honest about which ones you actually know versus which "look familiar." Paste your identity list into FlashDeck and it turns each one into a card from exactly what you pasted. Tap Knew it or Missed it, and misses come back tomorrow while the ones you know space out: Build my study plan.

Practice verifying identities

Memorizing is step one. The test usually asks you to verify or prove an identity. Some reliable moves:

1. Work on the messier side. Simplify it until it matches the cleaner side.
2. Convert everything to sin and cos. When stuck, this almost always gets you moving.
3. Look for a Pythagorean swap. See 1 − sin²? That's cos². See sec² − 1? That's tan².
4. Combine fractions over a common denominator.
5. Multiply by a conjugate when you see 1 ± sin θ or 1 ± cos θ in a denominator.
6. Never move terms across the equals sign. You're showing one side becomes the other, not solving an equation.

Mini example

Verify: sec θ − cos θ = sin θ tan θ

Left side: 1/cos θ − cos θ = (1 − cos²θ) / cos θ = sin²θ / cos θ = sin θ · (sin θ / cos θ) = sin θ tan θ. (checks out) Done.

Do three to five of these a day in the week before your test. It's the best way to make the identities feel usable, not just memorized.

Know which identity to reach for, instantly

Paste your trig identities into FlashDeck and in seconds you have one card per identity. About ten minutes a day, it shows only the ones you're about to forget and tracks what's learned, due and missed, so on test day the core identities are automatic and your brain is free for the proof.

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FAQ

What trig identities should you memorize?

Memorize the reciprocal and quotient definitions, sin²θ + cos²θ = 1, and the sine and cosine sum and difference formulas. From those you can derive the other Pythagorean identities, double-angle formulas and power-reducing formulas. Check your exam's formula sheet for anything provided.

What is the easiest way to remember trig identities?

Learn a tiny core and derive the rest, using hooks like "sine mixes and keeps the sign, cosine matches and flips it." Then review them with flashcards a few minutes a day so the common ones become automatic.

How do I memorize all trig identities for a test?

You don't need to memorize every identity. Learn the core ones cold, practice deriving the others quickly, and drill the ones you use most often. Doing verification problems daily is what makes them stick.

How do you verify trig identities?

Start with the more complicated side, convert to sine and cosine, and use Pythagorean swaps and common denominators to simplify it until it matches the other side. Don't move terms across the equals sign.

Is there a trig identities cheat sheet I can use?

A formula sheet is great for studying, but use it to make flashcards and quiz yourself rather than just reading it. Rereading a list builds recognition; recalling it builds memory.