Trig Identities Cheat Sheet: Every Formula You Need in One Place

Trigonometric identities are equations that are true for all values of the variable. They are the backbone of simplifying expressions, solving equations, and proving statements in trigonometry. This cheat sheet organizes every identity you need into clean reference tables, with memorization tips and worked examples.

If you are new to trigonometry or need a broader overview, start with our Trigonometry Guide.

Reciprocal Identities

These define the three secondary trig functions in terms of the primary ones.

Identity	Equivalent
csc(x)	1 / sin(x)
sec(x)	1 / cos(x)
cot(x)	1 / tan(x)

Memorization tip: Each pair shares a letter — sin and sec are NOT partners. Sin pairs with csc (co-secant), cos pairs with sec, tan pairs with cot. The “co-” prefix swaps columns.

Quotient Identities

Identity	Formula
tan(x)	sin(x) / cos(x)
cot(x)	cos(x) / sin(x)

These two come up constantly when simplifying. Anytime you see tan or cot, consider rewriting in terms of sin and cos.

Pythagorean Identities

These are arguably the most important identities in all of trigonometry.

Identity
sin²(x) + cos²(x) = 1
1 + tan²(x) = sec²(x)
1 + cot²(x) = csc²(x)

Memorization tip: You only need to memorize the first one. Divide both sides by cos²(x) to get the second, or by sin²(x) to get the third.

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Sum and Difference Formulas

Identity	Formula
sin(A + B)	sin(A)cos(B) + cos(A)sin(B)
sin(A - B)	sin(A)cos(B) - cos(A)sin(B)
cos(A + B)	cos(A)cos(B) - sin(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))
tan(A - B)	(tan(A) - tan(B)) / (1 + tan(A)tan(B))

Memorization tip: For sine, the sign in the middle matches the sign in the argument. For cosine, it flips. “Sine is friendly, cosine is contrary.”

Double Angle Formulas

Set A = B in the sum formulas and you get these.

Identity	Formula
sin(2x)	2 sin(x) cos(x)
cos(2x)	cos²(x) - sin²(x)
cos(2x)	2cos²(x) - 1
cos(2x)	1 - 2sin²(x)
tan(2x)	2tan(x) / (1 - tan²(x))

The three forms of cos(2x) are all useful. Pick the one that matches whatever variable you are trying to isolate.

Half Angle Formulas

Identity	Formula
sin(x/2)	plus or minus sqrt((1 - cos(x)) / 2)
cos(x/2)	plus or minus sqrt((1 + cos(x)) / 2)
tan(x/2)	sin(x) / (1 + cos(x))
tan(x/2)	(1 - cos(x)) / sin(x)

The plus-or-minus sign depends on which quadrant x/2 falls in.

Cofunction Identities

Identity	Equivalent
sin(pi/2 - x)	cos(x)
cos(pi/2 - x)	sin(x)
tan(pi/2 - x)	cot(x)
cot(pi/2 - x)	tan(x)
sec(pi/2 - x)	csc(x)
csc(pi/2 - x)	sec(x)

Memorization tip: “Co-function” literally means “complement function.” Two angles are complements when they add to 90 degrees (pi/2). So the cosine of an angle equals the sine of its complement.

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Worked Examples

Example 1: Simplify sin²(x) + sin²(x)tan²(x)

Factor out sin²(x):

sin²(x)(1 + tan²(x))

Apply the Pythagorean identity 1 + tan²(x) = sec²(x):

sin²(x) sec²(x)

Rewrite sec²(x) as 1/cos²(x):

sin²(x) / cos²(x) = tan²(x)

Final answer: tan²(x)

Example 2: Prove that (1 - cos(2x)) / sin(2x) = tan(x)

Use double angle formulas. Replace cos(2x) with 1 - 2sin²(x) and sin(2x) with 2sin(x)cos(x):

(1 - (1 - 2sin²(x))) / (2sin(x)cos(x))

Simplify the numerator:

2sin²(x) / (2sin(x)cos(x))

Cancel 2sin(x):

sin(x) / cos(x) = tan(x)

The identity is proven.

Example 3: Find the exact value of cos(75 degrees)

Write 75 as 45 + 30 and apply the cosine sum formula:

cos(75) = cos(45)cos(30) - sin(45)sin(30)

Substitute known values:

cos(75) = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2)

cos(75) = sqrt(6)/4 - sqrt(2)/4

Final answer: (sqrt(6) - sqrt(2)) / 4

For more on how these identities connect to calculus, see our guide on derivatives of trig functions.

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General Strategy for Using Trig Identities

1. Convert everything to sin and cos. This is the single most reliable first step.
2. Look for Pythagorean identity opportunities. If you see sin² + cos², 1 - sin², sec² - 1, etc., substitute.
3. Factor when possible. Common factors hide simplifications.
4. Work on the more complicated side. When proving an identity, start with whichever side looks messier and try to simplify it toward the other.

FAQ

How many trig identities do I actually need to memorize?

Focus on the Pythagorean identities, quotient identities, and sum/difference formulas. Everything else (double angle, half angle, cofunction) can be derived from those. In practice, memorizing the double angle formulas as well saves a lot of time on exams.

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

An identity is true for all valid values of the variable. An equation is true only for specific values. For example, sin²(x) + cos²(x) = 1 is an identity. sin(x) = 1/2 is an equation with specific solutions.

How do I prove a trig identity?

Start with one side (usually the more complex one) and use algebraic manipulation and known identities to transform it into the other side. Never move terms across the equals sign as if solving an equation.

Why do trig identities matter in calculus?

Many integrals and derivatives require rewriting trig expressions into equivalent forms before you can apply standard rules. For example, integrating sin²(x) requires the double angle identity cos(2x) = 1 - 2sin²(x) to rewrite it as (1 - cos(2x))/2.