What is the most important trig identity to memorize?

The most important trig identity is sin²θ + cos²θ = 1. It is the only one you truly need to memorize because the other Pythagorean identities (1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ) can be derived from it in one step by dividing through by cos²θ or sin²θ.

What is the difference between a trig identity and a trig equation?

A trig identity is true for all values of the variable where both sides are defined — for example, sin²θ + cos²θ = 1 holds for every angle. A trig equation is only true for specific angle values and must be solved — for example, sin θ = 0.5 is only true at θ = 30° and 150° within one period.

How do you use trig identities to simplify expressions?

Work on one side of the expression only — never move terms across the equal sign. Substitute a known identity to replace a complex term with a simpler one, factor or expand as needed, and repeat until both sides match. Starting with the more complicated side usually leads to a solution faster.

Are co-function identities tested on the ACT and SAT?

Co-function identities can appear on both the ACT and SAT, typically as part of a problem asking you to rewrite sin(90° − θ) as cosθ or vice versa. They are more frequently tested on AP Precalculus and AP Calculus exams, where simplifying expressions using complementary angle relationships is a common task.

How many trig identities do you need to know for AP exams?

For AP Precalculus and AP Calculus AB, you should know the Pythagorean identities, reciprocal identities, sum and difference formulas, double angle formulas, and the power-reduction forms derived from them. Half angle formulas and product-to-sum formulas appear more in BC and college-level courses.

Trig Identities Cheat Sheet: Every Formula You Need (With Examples)

This trig identities cheat sheet covers all eight families of trigonometric identities in a single reference page — organized for quick lookup during a study session or exam prep. Use the quick-reference table to scan all identities at once, then jump to any section for a worked example and plain-English explanation.

Trig identities matter because they let you simplify complex expressions, solve equations that resist direct methods, and are tested in AP Precalculus, AP Calculus AB/BC, and introductory college trig. The trig functions calculator can verify any sine, cosine, or tangent value instantly.

Formulas verified against standard US trig and AP Precalculus curricula. Last reviewed September 2026.

Quick-Reference Table: All Trig Identities at a Glance

Identity Family	Formula(s)	Exam note
Pythagorean	sin²θ + cos²θ = 1 1 + tan²θ = sec²θ 1 + cot²θ = csc²θ	Memorize all three
Reciprocal	csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ	Used constantly in simplification
Quotient	tan θ = sin θ/cos θ, cot θ = cos θ/sin θ	Foundation of any tan/cot simplification
Co-function	sin(90°−θ) = cos θ, cos(90°−θ) = sin θ, tan(90°−θ) = cot θ	Degrees and radians both appear on exams
Even/Odd	cos(−θ) = cos θ; sin(−θ) = −sin θ; tan(−θ) = −tan θ	Used when simplifying negative-angle expressions
Sum & Difference	sin(α±β) = sinα cosβ ± cosα sinβ cos(α±β) = cosα cosβ ∓ sinα sinβ	Most commonly tested identity family
Double Angle	sin 2θ = 2 sinθ cosθ cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ	Three cos variants — know when to use each
Half Angle	sin(θ/2) = ±√((1−cosθ)/2) cos(θ/2) = ±√((1+cosθ)/2)	Sign depends on quadrant of θ/2

Pythagorean Identities

The Three Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

How to Derive the Secant and Cosecant Variants

You only need to memorize the first identity. The other two follow in one step:

Divide sin²θ + cos²θ = 1 through by cos²θ to get: tan²θ + 1 = sec²θ
Divide sin²θ + cos²θ = 1 through by sin²θ to get: 1 + cot²θ = csc²θ

Common mistake: Writing (sinθ + cosθ)² = 1. This is wrong. The correct identity is sin²θ + cos²θ = 1, which means the squares of sin and cos sum to 1, not sin and cos themselves.

Use Pythagorean identities when: your expression contains sin²θ or cos²θ and you need to substitute one for the other to simplify.

See right triangle side relationships with the Pythagorean theorem calculator.

Reciprocal Identities

Function	Reciprocal identity
csc θ	1 / sin θ
sec θ	1 / cos θ
cot θ	1 / tan θ (= cos θ / sin θ)

These identities appear constantly when simplifying expressions or proving other identities. If you see csc, sec, or cot in a problem, the first step is often to convert using these reciprocal definitions. Verify any trig function value with the trig functions calculator.

Quotient Identities

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

These connect directly to SOH-CAH-TOA: tan = opposite/adjacent = (opposite/hypotenuse) ÷ (adjacent/hypotenuse) = sinθ/cosθ. Any time you see an expression with tan or cot, converting to sin/cos using these identities is the standard first step toward simplification.

Co-function Identities

Co-function identities describe the relationship between a trig function of an angle and the trig function of its complement (90° − θ or π/2 − θ).

Function	Co-function identity (degrees)	Co-function identity (radians)
sin θ	sin(90° − θ) = cos θ	sin(π/2 − θ) = cos θ
cos θ	cos(90° − θ) = sin θ	cos(π/2 − θ) = sin θ
tan θ	tan(90° − θ) = cot θ	tan(π/2 − θ) = cot θ
csc θ	csc(90° − θ) = sec θ	csc(π/2 − θ) = sec θ
sec θ	sec(90° − θ) = csc θ	sec(π/2 − θ) = csc θ
cot θ	cot(90° − θ) = tan θ	cot(π/2 − θ) = tan θ

Both degree and radian forms appear on exams. The pattern: any trig function of an angle equals the co-function of its complement.

Even and Odd Identities

Whether a trig function is even or odd determines how it behaves with negative angles.

Even functions are symmetric about the y-axis: f(−θ) = f(θ)
Odd functions are symmetric about the origin: f(−θ) = −f(θ)

Identity	Even or odd?
cos(−θ) = cos θ	Even
sec(−θ) = sec θ	Even
sin(−θ) = −sin θ	Odd
tan(−θ) = −tan θ	Odd
csc(−θ) = −csc θ	Odd
cot(−θ) = −cot θ	Odd

Use even/odd identities when: you encounter a negative angle or a subtracted angle argument. Replace the negative angle with the positive-angle form and adjust the sign according to even/odd status.

Sum and Difference Identities

Sum and Difference Formulas for Sine and Cosine

sin(α + β) = sin α cos β + cos α sin β

sin(α − β) = sin α cos β − cos α sin β

cos(α + β) = cos α cos β − sin α sin β

cos(α − β) = cos α cos β + sin α sin β

Common mistake: The cosine difference formula has a plus sign: cos(α − β) = cosα cosβ + sinα sinβ. Students often flip this sign.

Sum and Difference Formulas for Tangent

tan(α + β) = (tan α + tan β) / (1 − tan α tan β)

tan(α − β) = (tan α − tan β) / (1 + tan α tan β)

Worked Example: Exact Value of sin(75°)

Write 75° = 45° + 30°, then apply the sum formula:

sin(75°) = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2) / 4

Double Angle Identities

Double Angle Formulas for Sine and Tangent

sin 2θ = 2 sin θ cos θ

tan 2θ = (2 tan θ) / (1 − tan²θ)

Three Forms of the Cosine Double Angle Formula

All three are equivalent — derived from cos(α + β) with α = β = θ:

cos 2θ = cos²θ − sin²θ

cos 2θ = 2cos²θ − 1

cos 2θ = 1 − 2sin²θ

Which cosine double angle form to use: Use 1 − 2sin²θ when only sin is present in your expression. Use 2cos²θ − 1 when only cos is present. Use cos²θ − sin²θ when both appear.

Common mistake: Writing sin(2θ) = 2sin(θ). The double angle formula requires a product: sin 2θ = 2 sinθ cosθ.

Power-Reduction Identities (Derived from Double Angle)

These appear frequently in calculus integration:

sin²θ = (1 − cos 2θ) / 2

cos²θ = (1 + cos 2θ) / 2

Half Angle Identities

Half Angle Formulas for Sine, Cosine, and Tangent

sin(θ/2) = ±√((1 − cos θ) / 2)

cos(θ/2) = ±√((1 + cos θ) / 2)

tan(θ/2) = ±√((1 − cos θ) / (1 + cos θ)) = sin θ / (1 + cos θ) = (1 − cos θ) / sin θ

How to Determine the Correct ± Sign

This is the step that trips students up on exams. The ± is not ambiguous — it is determined by the quadrant that θ/2 falls in:

Find the value of θ/2 (half the given angle).
Determine which quadrant θ/2 is in.
Apply ASTC (All Students Take Calculus): Quadrant I — all positive; Quadrant II — only sin positive; Quadrant III — only tan positive; Quadrant IV — only cos positive.
Assign + or − based on the sign of the function you are calculating in that quadrant.

Worked Example: Exact Value of sin(22.5°)

22.5° = 45°/2, so θ = 45°. Since 22.5° is in Quadrant I, sin is positive.

sin(22.5°) = √((1 − cos 45°) / 2)
cos 45° = √2/2
= √((1 − √2/2) / 2) = √((2 − √2) / 4)
= √(2 − √2) / 2

Which Trig Identity Should You Use? A Quick Decision Guide

Knowing the formulas is step one — knowing which to apply is step two. Use these if/then rules:

Expression contains sin²θ or cos²θ? → Start with a Pythagorean identity to substitute one for the other.
Need an exact value at a non-standard angle (e.g., 75°, 15°, 195°)? → Write the angle as a sum or difference of standard angles (30°, 45°, 60°, 90°), then use a sum/difference identity.
Expression has 2θ (double angle)? → Use a double angle identity. Pick the cosine version based on what's already in the expression.
Expression has θ/2 (half angle)? → Use a half angle identity. Always check the quadrant of θ/2 to assign the sign.
Expression has a negative angle like sin(−θ)? → Use an even/odd identity to rewrite it with a positive angle.

How to Verify a Trig Identity (Step-by-Step)

The 5-Step Verification Method

Work on one side only — never move terms across the equal sign.
Convert all functions to sin and cos (use reciprocal and quotient identities).
Apply a Pythagorean identity if any sin² or cos² terms appear.
Factor, expand, or combine fractions as needed.
Keep simplifying until your side matches the other side exactly.

Worked Example: Verify sin²θ / (1 − cos θ) = 1 + cos θ

Work on the left side: sin²θ / (1 − cosθ)
Replace sin²θ with (1 − cos²θ) using the Pythagorean identity: (1 − cos²θ) / (1 − cosθ)
Factor the numerator as a difference of squares: (1 − cosθ)(1 + cosθ) / (1 − cosθ)
Cancel (1 − cosθ) from numerator and denominator: 1 + cosθ ✓

Common Mistakes to Avoid

Canceling sin from sin²θ + sinθ: You cannot factor out and cancel sin from a sum. Factor out sinθ first: sinθ(sinθ + 1).
Assuming sin(α + β) = sinα + sinβ: This is wrong. The sum formula has four terms: sinα cosβ + cosα sinβ.
Forgetting the ± in half-angle formulas: The sign is determined by the quadrant of θ/2, not θ.
Mixing up the three cosine double-angle variants: All three are correct. The exam problem tells you which one to use based on what's already in the expression.
Dividing both sides by sinθ or cosθ: This is only valid when sinθ ≠ 0 or cosθ ≠ 0. Dividing can eliminate solutions.
Working both sides toward the middle: Always work on one side only. Moving terms from both sides is not a valid proof technique.

Frequently Asked Questions

What is the most important trig identity to memorize?

The most important trig identity is sin²θ + cos²θ = 1. It is the only one you truly need to memorize because the other Pythagorean identities can be derived from it in one step by dividing through by cos²θ or sin²θ.
What is the difference between a trig identity and a trig equation?

A trig identity is true for all values of the variable where both sides are defined — sin²θ + cos²θ = 1 holds for every angle. A trig equation is only true for specific angle values and must be solved — for example, sin θ = 0.5 is only true at θ = 30° and 150° within one period.
How do you use trig identities to simplify expressions?

Work on one side of the expression only — never move terms across the equal sign. Substitute a known identity to replace a complex term with a simpler one, factor or expand as needed, and repeat until both sides match. Starting with the more complicated side usually leads to a solution faster.
Are co-function identities tested on the ACT and SAT?

Co-function identities can appear on both the ACT and SAT as part of a problem asking you to rewrite sin(90° − θ) as cosθ. They are more frequently tested on AP Precalculus and AP Calculus exams, where simplifying complementary angle expressions is a standard task.
How many trig identities do you need to know for AP exams?

For AP Precalculus and AP Calculus AB, focus on the Pythagorean identities, reciprocal identities, sum and difference formulas, and double angle formulas including all three cosine variants. Half angle formulas and power-reduction forms appear more in BC and college-level courses.

Use the trig functions calculator to verify any sine, cosine, or tangent value — step-by-step, no login required. For triangle problems, try the law of sines calculator or the law of cosines calculator. Studying for AP Precalculus? See which of these identities are actually tested. Explore all 27 free math calculators at MathInSite.

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