What are trigonometric identities?

Trigonometric identities are equations involving trig functions that hold for every valid value of the variable. They cover Pythagorean relations such as sin squared plus cos squared equals one, reciprocal and quotient definitions, even and odd behavior, cofunction relations, sum-angle and double-angle formulas, half-angle and power-reduction forms, product-to-sum conversions, reference-angle reduction, and inverse-function relationships. Together they form the working toolkit for simplifying, integrating, and solving trig expressions, covering all six functions: sine, cosine, tangent, cotangent, secant, and cosecant.

What is the Pythagorean identity?

The fundamental Pythagorean identity is sin squared of x plus cos squared of x equals one, true for every x. It follows from the unit circle: any point on it has coordinates (cos theta, sin theta) and satisfies x squared plus y squared equals one. Two derived forms follow by dividing through by cos squared or sin squared: one plus tan squared equals sec squared, and one plus cot squared equals csc squared.

What are the trigonometric reduction formulas?

Reduction formulas, also called reference-angle reduction, let you rewrite any trig function evaluated at any angle as the same function (or its cofunction) evaluated at a first-quadrant angle, up to a sign. The standard transformations are pi over two plus or minus x, pi plus or minus x, three pi over two plus or minus x, and two pi plus or minus x. For example, sin(pi + x) equals minus sin(x), and cos(three pi over two minus x) equals minus sin(x). They cover all six functions and are essential for evaluating trig at any angle in terms of acute-angle values.

What is the difference between sum-angle and double-angle identities?

Sum-angle identities give sin(a + b), cos(a + b), and tan(a + b) in terms of trig functions of a and b separately. Double-angle identities are the special case where a equals b equals x, producing sin(2x) equals 2 sin x cos x, cos(2x) equals cos squared minus sin squared, and so on. Sum-angle is the general form; double-angle is the most commonly used specialization, and triple-angle, half-angle, and power-reduction all descend from it by substitution or algebraic rearrangement. All six functions (sine, cosine, tangent, cotangent, secant, cosecant) have versions of each.

How do half-angle and power-reduction formulas relate?

They are two sides of the same algebraic move. Power-reduction starts from the double-angle form cos(2x) equals one minus two sin squared x, solved for sin squared x to give one half of (one minus cos 2x). Half-angle takes the square root of the power-reduction form, giving sin(x over two) equals plus-or-minus the square root of (one minus cos x) over two. Power-reduction reduces a squared power to a multiple-angle expression; half-angle reduces a half angle to a full-angle expression.

How do I memorize trig identities?

Group them by family rather than memorizing in isolation. Start from the unit circle (Pythagorean), then reciprocal and quotient definitions. Master sum-angle for sine and cosine — everything else (difference, double, triple, half, product-to-sum, sum-to-product) follows by substitution. Note the parities: cosine and secant are even, while sine, tangent, cotangent, and cosecant are odd. Learn the reference-angle reduction transforms (pi over two plus or minus x, pi plus or minus x, and so on) to reduce any angle to a first-quadrant equivalent. Drill the family relationships with the drag puzzle on this page, and use the search-by-LHS feature to verify forms quickly.

Trigonometric Identities

Reference table of trigonometric identities. Try puzzle mode to drill, or read the full trig identities explanation →

Trig identities tool

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Pythagorean

\sin^2(x) + \cos^2(x)

Pythagorean identity

↓

1

Pythagorean

1 + \tan^2(x)

Pythagorean (tan / sec)

↓

\sec^2(x)

Pythagorean

1 + \cot^2(x)

Pythagorean (cot / csc)

↓

\csc^2(x)

Reciprocal & quotient

\csc(x)

Cosecant as reciprocal

↓

\dfrac{1}{\sin(x)}

Reciprocal & quotient

\sec(x)

Secant as reciprocal

↓

\dfrac{1}{\cos(x)}

Reciprocal & quotient

\cot(x)

Cotangent as reciprocal of tangent

↓

\dfrac{1}{\tan(x)}

Reciprocal & quotient

\tan(x)

Tangent as quotient

↓

\dfrac{\sin(x)}{\cos(x)}

Reciprocal & quotient

\cot(x)

Cotangent as quotient

↓

\dfrac{\cos(x)}{\sin(x)}

Negative-angle

\sin(-x)

Sine is odd

↓

-\sin(x)

Negative-angle

\cos(-x)

Cosine is even

↓

\cos(x)

Negative-angle

\tan(-x)

Tangent is odd

↓

-\tan(x)

Negative-angle

\cot(-x)

Cotangent is odd

↓

-\cot(x)

Negative-angle

\sec(-x)

Secant is even

↓

\sec(x)

Negative-angle

\csc(-x)

Cosecant is odd

↓

-\csc(x)

Complement (cofunction)

\sin\!\left(\tfrac{\pi}{2} - x\right)

Sine of complement = cosine

↓

\cos(x)

Complement (cofunction)

\cos\!\left(\tfrac{\pi}{2} - x\right)

Cosine of complement = sine

↓

\sin(x)

Complement (cofunction)

\tan\!\left(\tfrac{\pi}{2} - x\right)

Tangent of complement = cotangent

↓

\cot(x)

Complement (cofunction)

\cot\!\left(\tfrac{\pi}{2} - x\right)

Cotangent of complement = tangent

↓

\tan(x)

Complement (cofunction)

\sec\!\left(\tfrac{\pi}{2} - x\right)

Secant of complement = cosecant

↓

\csc(x)

Complement (cofunction)

\csc\!\left(\tfrac{\pi}{2} - x\right)

Cosecant of complement = secant

↓

\sec(x)

Supplement

\sin(\pi - x)

Sine of supplement

↓

\sin(x)

Supplement

\cos(\pi - x)

Cosine of supplement

↓

-\cos(x)

Supplement

\tan(\pi - x)

Tangent of supplement

↓

-\tan(x)

Supplement

\cot(\pi - x)

Cotangent of supplement

↓

-\cot(x)

Supplement

\sec(\pi - x)

Secant of supplement

↓

-\sec(x)

Supplement

\csc(\pi - x)

Cosecant of supplement

↓

\csc(x)

Reference-angle reduction

\sin\!\left(\tfrac{\pi}{2} + x\right)

Sine of π/2 + x

↓

\cos(x)

Reference-angle reduction

\cos\!\left(\tfrac{\pi}{2} + x\right)

Cosine of π/2 + x

↓

-\sin(x)

Reference-angle reduction

\tan\!\left(\tfrac{\pi}{2} + x\right)

Tangent of π/2 + x

↓

-\cot(x)

Reference-angle reduction

\cot\!\left(\tfrac{\pi}{2} + x\right)

Cotangent of π/2 + x

↓

-\tan(x)

Reference-angle reduction

\sec\!\left(\tfrac{\pi}{2} + x\right)

Secant of π/2 + x

↓

-\csc(x)

Reference-angle reduction

\csc\!\left(\tfrac{\pi}{2} + x\right)

Cosecant of π/2 + x

↓

\sec(x)

Reference-angle reduction

\sin(\pi + x)

Sine of π + x

↓

-\sin(x)

Reference-angle reduction

\cos(\pi + x)

Cosine of π + x

↓

-\cos(x)

Reference-angle reduction

\tan(\pi + x)

Tangent of π + x

↓

\tan(x)

Reference-angle reduction

\cot(\pi + x)

Cotangent of π + x

↓

\cot(x)

Reference-angle reduction

\sec(\pi + x)

Secant of π + x

↓

-\sec(x)

Reference-angle reduction

\csc(\pi + x)

Cosecant of π + x

↓

-\csc(x)

Reference-angle reduction

\sin\!\left(\tfrac{3\pi}{2} - x\right)

Sine of 3π/2 − x

↓

-\cos(x)

Reference-angle reduction

\cos\!\left(\tfrac{3\pi}{2} - x\right)

Cosine of 3π/2 − x

↓

-\sin(x)

Reference-angle reduction

\tan\!\left(\tfrac{3\pi}{2} - x\right)

Tangent of 3π/2 − x

↓

\cot(x)

Reference-angle reduction

\cot\!\left(\tfrac{3\pi}{2} - x\right)

Cotangent of 3π/2 − x

↓

\tan(x)

Reference-angle reduction

\sec\!\left(\tfrac{3\pi}{2} - x\right)

Secant of 3π/2 − x

↓

-\csc(x)

Reference-angle reduction

\csc\!\left(\tfrac{3\pi}{2} - x\right)

Cosecant of 3π/2 − x

↓

-\sec(x)

Reference-angle reduction

\sin\!\left(\tfrac{3\pi}{2} + x\right)

Sine of 3π/2 + x

↓

-\cos(x)

Reference-angle reduction

\cos\!\left(\tfrac{3\pi}{2} + x\right)

Cosine of 3π/2 + x

↓

\sin(x)

Reference-angle reduction

\tan\!\left(\tfrac{3\pi}{2} + x\right)

Tangent of 3π/2 + x

↓

-\cot(x)

Reference-angle reduction

\cot\!\left(\tfrac{3\pi}{2} + x\right)

Cotangent of 3π/2 + x

↓

-\tan(x)

Reference-angle reduction

\sec\!\left(\tfrac{3\pi}{2} + x\right)

Secant of 3π/2 + x

↓

\csc(x)

Reference-angle reduction

\csc\!\left(\tfrac{3\pi}{2} + x\right)

Cosecant of 3π/2 + x

↓

-\sec(x)

Reference-angle reduction

\sin(2\pi - x)

Sine of 2π − x

↓

-\sin(x)

Reference-angle reduction

\cos(2\pi - x)

Cosine of 2π − x

↓

\cos(x)

Reference-angle reduction

\tan(2\pi - x)

Tangent of 2π − x

↓

-\tan(x)

Reference-angle reduction

\cot(2\pi - x)

Cotangent of 2π − x

↓

-\cot(x)

Reference-angle reduction

\sec(2\pi - x)

Secant of 2π − x

↓

\sec(x)

Reference-angle reduction

\csc(2\pi - x)

Cosecant of 2π − x

↓

-\csc(x)

Reference-angle reduction

\sin(2\pi + x)

Sine of 2π + x

↓

\sin(x)

Reference-angle reduction

\cos(2\pi + x)

Cosine of 2π + x

↓

\cos(x)

Reference-angle reduction

\tan(2\pi + x)

Tangent of 2π + x

↓

\tan(x)

Reference-angle reduction

\cot(2\pi + x)

Cotangent of 2π + x

↓

\cot(x)

Reference-angle reduction

\sec(2\pi + x)

Secant of 2π + x

↓

\sec(x)

Reference-angle reduction

\csc(2\pi + x)

Cosecant of 2π + x

↓

\csc(x)

Sum-angle

\sin(a + b)

Sine of a sum

↓

\sin(a)\cos(b) + \cos(a)\sin(b)

Sum-angle

\cos(a + b)

Cosine of a sum

↓

\cos(a)\cos(b) - \sin(a)\sin(b)

Sum-angle

\tan(a + b)

Tangent of a sum

↓

\dfrac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}

Sum-angle

\cot(a + b)

Cotangent of a sum

↓

\dfrac{\cot(a)\cot(b) - 1}{\cot(a) + \cot(b)}

Sum-angle

\sec(a + b)

Secant of a sum

↓

\dfrac{\sec(a)\sec(b)}{1 - \tan(a)\tan(b)}

Sum-angle

\csc(a + b)

Cosecant of a sum

↓

\dfrac{\csc(a)\csc(b)}{\cot(a) + \cot(b)}

Difference-angle

\sin(a - b)

Sine of a difference

↓

\sin(a)\cos(b) - \cos(a)\sin(b)

Difference-angle

\cos(a - b)

Cosine of a difference

↓

\cos(a)\cos(b) + \sin(a)\sin(b)

Difference-angle

\tan(a - b)

Tangent of a difference

↓

\dfrac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}

Difference-angle

\cot(a - b)

Cotangent of a difference

↓

\dfrac{\cot(a)\cot(b) + 1}{\cot(b) - \cot(a)}

Difference-angle

\sec(a - b)

Secant of a difference

↓

\dfrac{\sec(a)\sec(b)}{1 + \tan(a)\tan(b)}

Difference-angle

\csc(a - b)

Cosecant of a difference

↓

\dfrac{\csc(a)\csc(b)}{\cot(b) - \cot(a)}

Double-angle

\sin(2x)

Sine double-angle

↓

2\sin(x)\cos(x)

Double-angle

\cos(2x)

Cosine double-angle

↓

\cos^2(x) - \sin^2(x)

Double-angle

\tan(2x)

Tangent double-angle

↓

\dfrac{2\tan(x)}{1 - \tan^2(x)}

Double-angle

\cot(2x)

Cotangent double-angle

↓

\dfrac{\cot^2(x) - 1}{2\cot(x)}

Double-angle

\sec(2x)

Secant double-angle

↓

\dfrac{\sec^2(x)}{2 - \sec^2(x)}

Double-angle

\csc(2x)

Cosecant double-angle

↓

\dfrac{\sec(x)\csc(x)}{2}

Triple-angle

\sin(3x)

Sine triple-angle

↓

3\sin(x) - 4\sin^3(x)

Triple-angle

\cos(3x)

Cosine triple-angle

↓

4\cos^3(x) - 3\cos(x)

Triple-angle

\tan(3x)

Tangent triple-angle

↓

\dfrac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)}

Triple-angle

\cot(3x)

Cotangent triple-angle

↓

\dfrac{\cot^3(x) - 3\cot(x)}{3\cot^2(x) - 1}

Triple-angle

\sec(3x)

Secant triple-angle

↓

\dfrac{\sec(x)}{4\cos^2(x) - 3}

Triple-angle

\csc(3x)

Cosecant triple-angle

↓

\dfrac{\csc(x)}{3 - 4\sin^2(x)}

Half-angle

\sin\!\left(\tfrac{x}{2}\right)

Sine half-angle

↓

\pm\sqrt{\dfrac{1 - \cos(x)}{2}}

Half-angle

\cos\!\left(\tfrac{x}{2}\right)

Cosine half-angle

↓

\pm\sqrt{\dfrac{1 + \cos(x)}{2}}

Half-angle

\tan\!\left(\tfrac{x}{2}\right)

Tangent half-angle

↓

\dfrac{1 - \cos(x)}{\sin(x)}

Half-angle

\cot\!\left(\tfrac{x}{2}\right)

Cotangent half-angle

↓

\dfrac{1 + \cos(x)}{\sin(x)}

Half-angle

\sec\!\left(\tfrac{x}{2}\right)

Secant half-angle

↓

\pm\sqrt{\dfrac{2}{1 + \cos(x)}}

Half-angle

\csc\!\left(\tfrac{x}{2}\right)

Cosecant half-angle

↓

\pm\sqrt{\dfrac{2}{1 - \cos(x)}}

Power reduction

\sin^2(x)

Sine squared

↓

\dfrac{1 - \cos(2x)}{2}

Power reduction

\cos^2(x)

Cosine squared

↓

\dfrac{1 + \cos(2x)}{2}

Power reduction

\tan^2(x)

Tangent squared

↓

\dfrac{1 - \cos(2x)}{1 + \cos(2x)}

Product-to-sum

\sin(a)\cos(b)

Sine times cosine

↓

\tfrac{1}{2}\bigl[\sin(a + b) + \sin(a - b)\bigr]

Product-to-sum

\cos(a)\sin(b)

Cosine times sine

↓

\tfrac{1}{2}\bigl[\sin(a + b) - \sin(a - b)\bigr]

Product-to-sum

\cos(a)\cos(b)

Cosine times cosine

↓

\tfrac{1}{2}\bigl[\cos(a - b) + \cos(a + b)\bigr]

Product-to-sum

\sin(a)\sin(b)

Sine times sine

↓

\tfrac{1}{2}\bigl[\cos(a - b) - \cos(a + b)\bigr]

Sum-to-product

\sin(a) + \sin(b)

Sine plus sine

↓

2\sin\!\left(\tfrac{a + b}{2}\right)\cos\!\left(\tfrac{a - b}{2}\right)

Sum-to-product

\sin(a) - \sin(b)

Sine minus sine

↓

2\cos\!\left(\tfrac{a + b}{2}\right)\sin\!\left(\tfrac{a - b}{2}\right)

Sum-to-product

\cos(a) + \cos(b)

Cosine plus cosine

↓

2\cos\!\left(\tfrac{a + b}{2}\right)\cos\!\left(\tfrac{a - b}{2}\right)

Sum-to-product

\cos(a) - \cos(b)

Cosine minus cosine

↓

-2\sin\!\left(\tfrac{a + b}{2}\right)\sin\!\left(\tfrac{a - b}{2}\right)

Inverse

\sin(\arcsin x)

Sine of arcsine

↓

x

Inverse

\cos(\arccos x)

Cosine of arccosine

↓

x

Inverse

\tan(\arctan x)

Tangent of arctangent

↓

x

Inverse

\cos(\arcsin x)

Cosine of arcsine

↓

\sqrt{1 - x^2}

Inverse

\sin(\arccos x)

Sine of arccosine

↓

\sqrt{1 - x^2}

Inverse

\tan(\arcsin x)

Tangent of arcsine

↓

\dfrac{x}{\sqrt{1 - x^2}}

Inverse

\arcsin(x) + \arccos(x)

Arcsine plus arccosine

↓

\tfrac{\pi}{2}

Inverse

\arctan(x) + \operatorname{arccot}(x)

Arctangent plus arccotangent

↓

\tfrac{\pi}{2}

Inverse

\operatorname{arcsec}(x) + \operatorname{arccsc}(x)

Arcsecant plus arccosecant

↓

\tfrac{\pi}{2}

Inverse

\arcsin(-x)

Arcsine of negative

↓

-\arcsin(x)

Inverse

\arccos(-x)

Arccosine of negative

↓

\pi - \arccos(x)

Inverse

\arctan(-x)

Arctangent of negative

↓

-\arctan(x)

Families of identities

Click a family to highlight its entries in the table above.

sin²+cos²

Pythagorean

Identities that follow from the unit circle equation $x^2 + y^2 = 1$ .

3 matchesClick to highlight

1/sin

Reciprocal & quotient

Definitional identities expressing each trig function in terms of $\sin$ and $\cos$ .

5 matchesClick to highlight

f(-x)

Negative-angle

How each trig function responds to a sign flip on the input — the even/odd classification.

6 matchesClick to highlight

π/2 - x

Complement (cofunction)

Cofunction identities: each trig function equals its "co-" counterpart at the complementary angle.

6 matchesClick to highlight

π - x

Supplement

Identities relating trig functions of supplementary angles — reflection across the $y$ -axis on the unit circle.

6 matchesClick to highlight

π+x

Reference-angle reduction

Reduce any angle to a first-quadrant equivalent. Each transform (π/2±x, π±x, 3π/2±x, 2π±x) gives the trig function as a signed first-quadrant form.

36 matchesClick to highlight

a+b

Sum-angle

Identities for $\sin$ , $\cos$ , $\tan$ , $\cot$ , $\sec$ , $\csc$ of $a + b$ — the source from which most multi-angle identities derive.

6 matchesClick to highlight

a-b

Difference-angle

Identities for the six trig functions of $a - b$ . Recover by substituting $-b$ into the sum-angle formulas.

6 matchesClick to highlight

Double-angle

Identities for the six trig functions of $2x$ — the sum-angle identities with $a = b = x$ .

6 matchesClick to highlight

Triple-angle

Identities for the six trig functions of $3x$ — build by applying sum-angle to $(2x) + x$ .

6 matchesClick to highlight

x/2

Half-angle

Identities for the six trig functions at $x/2$ — derived from the power-reduction formulas by taking a square root.

6 matchesClick to highlight

sin²

Power reduction

Rewrite $\sin^2 x$ , $\cos^2 x$ , $\tan^2 x$ as expressions in $\cos(2x)$ — essential for integration of even powers.

3 matchesClick to highlight

×→+

Product-to-sum

Convert a product of trig functions into a sum or difference — the key trick for integrating products like $\sin(ax)\cos(bx)$ .

4 matchesClick to highlight

+→×

Sum-to-product

Convert a sum or difference of trig functions into a product. The inverse direction of product-to-sum, useful for factoring trig expressions.

4 matchesClick to highlight

sin⁻¹

Inverse

Identities for $\arcsin$ , $\arccos$ , $\arctan$ and friends — compositions, mixed compositions, cofunction sums, and negative-input behavior.

12 matchesClick to highlight

odd

Odd-function identities

Negative-angle identities for the odd trig functions: $\sin$ , $\tan$ , $\cot$ , $\csc$ .

4 matchesClick to highlight

even

Even-function identities

Negative-angle identities for the even trig functions: $\cos$ and $\sec$ .

2 matchesClick to highlight

How identities work

Foundational principles for understanding identity families.

≡

Identity vs. equation

A trigonometric identity holds for every valid value of the variable in its domain. A trigonometric equation, by contrast, holds only at specific values.

\sin^2(x) + \cos^2(x) = 1 \quad \text{(true for every } x \text{)}

Cofunction pairs

Each trig function pairs with a "co-" counterpart: sine with cosine, tangent with cotangent, secant with cosecant. Cofunctions of complementary angles are equal.

\sin\theta = \cos\!\left(\tfrac{\pi}{2} - \theta\right)

Even and odd

Cosine and secant are even. Sine, tangent, cotangent, and cosecant are odd. This determines how each function responds to a sign flip on the input.

\cos(-x) = \cos(x) \quad\text{(even)} \qquad \sin(-x) = -\sin(x) \quad\text{(odd)}

Building the Pythagorean family

A single identity $\sin^2 + \cos^2 = 1$ generates the family. Dividing through by $\cos^2(x)$ produces the tan-sec form; dividing by $\sin^2(x)$ produces the cot-csc form.

\dfrac{\sin^2 x}{\cos^2 x} + \dfrac{\cos^2 x}{\cos^2 x} = \dfrac{1}{\cos^2 x} \;\Longrightarrow\; \tan^2(x) + 1 = \sec^2(x)

Unit circle as the source

Most trig identities are restatements of geometric facts about the unit circle: the equation $x^2 + y^2 = 1$ , reflections across axes, and rotations.

(\cos\theta)^2 + (\sin\theta)^2 = 1 \quad \text{(the unit circle equation)}

↻

Reduction to a reference angle

Any trig function at any angle reduces to the same function (or its cofunction) at a first-quadrant angle, up to a sign. The transformations $\pi/2 \pm x$ , $\pi \pm x$ , $3\pi/2 \pm x$ , and $2\pi \pm x$ span every case. The sign comes from the quadrant; whether the function or its cofunction appears comes from whether the rotation is an odd or even multiple of $\pi/2$ .

\sin(\pi + x) = -\sin(x) \qquad \cos\!\left(\tfrac{3\pi}{2} - x\right) = -\sin(x)

→

Sum-angle as the generator

The sum-angle identities for $\sin$ and $\cos$ are the source of most multi-angle identities. Difference-angle follows by negating the second input; double-angle is the special case $a = b$ ; triple-angle is $\sin(2x + x)$ ; power-reduction comes from solving the double-angle form for the squared term; and half-angle comes from the power-reduction formulas. Product-to-sum follows by adding and subtracting sum-angle and difference-angle; sum-to-product follows by substituting $a = u + v$ , $b = u - v$ into product-to-sum.

\sin(2x) = \sin(x + x) = 2\sin(x)\cos(x)

Inverse trig and principal values

Inverse trig functions are not true inverses on the full real line — sine is not one-to-one, so $\arcsin$ requires restricting sine to $[-\tfrac{\pi}{2}, \tfrac{\pi}{2}]$ before inverting. The chosen restriction is called the principal value. Compositions like $\sin(\arcsin x)$ always recover $x$ ; the reverse direction $\arcsin(\sin x)$ only does so within the principal range.

\arcsin\!\left(\sin\tfrac{3\pi}{4}\right) = \arcsin\!\left(\tfrac{\sqrt 2}{2}\right) = \tfrac{\pi}{4} \;\neq\; \tfrac{3\pi}{4}