Precalculus Identities Reference Sheet

The Precalculus Identities Cheat Sheet You Actually Need

Stop scrolling through 47 tabs trying to find the right identity. This is the only precalculus identities reference sheet you'll need for your homework, tests, or that panic moment before class starts.

Every identity here is tested, accurate, and ready to use. No explanations of why math is beautiful. Just what works.

Fundamental Trigonometric Identities

These are the building blocks. If you forget everything else on this page, remember these six.

• sin(θ) = 1 / csc(θ)
• cos(θ) = 1 / sec(θ)
• tan(θ) = 1 / cot(θ)
• tan(θ) = sin(θ) / cos(θ)
• cot(θ) = cos(θ) / sin(θ)
• csc(θ) = 1 / sin(θ)

The reciprocal relationships will save you when you're stuck. If you know sine, you know cosecant. Same thing, inverted.

Pythagorean Identities

These come from the Pythagorean theorem applied to the unit circle. They come up constantly in simplification problems.

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = csc²(θ)

You can rearrange these to solve for any piece. Need sin²? It's 1 - cos². Need cos²? It's 1 - sin². Don't memorize the variations—memorize the base and solve.

Sum and Difference Formulas

These let you break apart angles that don't match your reference angles.

Sine Sum/Difference

sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

Cosine Sum/Difference

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Tangent Sum/Difference

tan(A + B) = [tan(A) + tan(B)] / [1 - tan(A)tan(B)]

tan(A - B) = [tan(A) - tan(B)] / [1 + tan(A)tan(B)]

Quick trick: sine keeps the sign between terms, cosine flips it, tangent has that denominator with the minus sign for adding.

Double Angle Identities

When you see 2θ, these are your weapons.

• sin(2θ) = 2sin(θ)cos(θ)
• cos(2θ) = cos²(θ) - sin²(θ)
• cos(2θ) = 2cos²(θ) - 1
• cos(2θ) = 1 - 2sin²(θ)
• tan(2θ) = 2tan(θ) / [1 - tan²(θ)]

For cosine, use whichever version matches what you're given. Working with sin²? Use 1 - 2sin²(θ). Working with cos²? Use 2cos²(θ) - 1.

Half Angle Identities

These are just the double angle formulas rearranged. They look messy but follow a pattern.

sin(θ/2) = ±√[(1 - cos(θ)) / 2]

cos(θ/2) = ±√[(1 + cos(θ)) / 2]

tan(θ/2) = ±√[(1 - cos(θ)) / (1 + cos(θ))]

The ± sign depends on which quadrant the angle lands in. Check your quadrant first.

Product-to-Sum Formulas

Convert products of trig functions into sums. Useful when you have messy products in integration or simplification problems.

• sin(A)cos(B) = ½[sin(A + B) + sin(A - B)]
• cos(A)sin(B) = ½[sin(A + B) - sin(A - B)]
• cos(A)cos(B) = ½[cos(A + B) + cos(A - B)]
• sin(A)sin(B) = ½[cos(A - B) - cos(A + B)]

Sum-to-Product Formulas

The reverse of above. Add trig functions instead of multiplying them.

• sin(A) + sin(B) = 2sin[(A + B)/2]cos[(A - B)/2]
• sin(A) - sin(B) = 2cos[(A + B)/2]sin[(A - B)/2]
• cos(A) + cos(B) = 2cos[(A + B)/2]cos[(A - B)/2]
• cos(A) - cos(B) = -2sin[(A + B)/2]sin[(A - B)/2]

Cofunction Identities

These show the relationship between trig functions of complementary angles (angles that add to π/2 or 90°).

Think of it this way: sine and cosine are cofunctions. Tangent and cotangent are cofunctions. Secant and cosecant are cofunctions.

Even and Odd Function Identities

These tell you what happens when you negate the angle.

• sin(-θ) = -sin(θ) — odd function
• cos(-θ) = cos(θ) — even function
• tan(-θ) = -tan(θ) — odd function

Only cosine and secant are even. Everything else flips sign with a negative angle.

Law of Sines and Cosines

These aren't technically identities, but you'll need them constantly in precalc and beyond.

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Law of Cosines

c² = a² + b² - 2ab·cos(C)

Use law of cosines when you have two sides and the included angle, or all three sides. Use law of sines when you have two angles and any side.

How to Use This Reference Sheet

For Simplifying Expressions

1. Look for patterns that match identities — sin² + cos², sum/difference structures, products that could become sums
2. Convert everything to sine and cosine first if stuck — reciprocals are easier to spot
3. Factor where possible before applying identities

For Verifying Identities

1. Start with the more complicated side
2. Convert everything to sine and cosine
3. Use Pythagorean identities to replace 1 when you see sin² + cos²
4. Factor, combine fractions, or use sum/difference formulas as needed
5. End when both sides match exactly

For Solving Equations

1. Use identities to get all trig functions with the same angle
2. Isolate the trig function
3. Solve for the angle using inverse functions
4. Find all solutions within the given interval

Common Mistakes That Cost Points

• Using the wrong sign — check your quadrant before picking ±
• Forgetting the 2 — sin(2θ) = 2sinθcosθ, not sinθcosθ
• Mixing up product-to-sum and sum-to-product — one has a 2, one has ½
• Applying Pythagorean identities wrong — tan² + 1 = sec², not sin² + cos²
• Ignoring the domain — tan and sec aren't defined at π/2 + kπ

Quick Reference Table

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