Trigonometric Identities

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

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Name	LHS		RHS	Family
Pythagorean identity	$\sin^2(x) + \cos^2(x)$	=	$1$	Pythagorean
Pythagorean (tan / sec)	$1 + \tan^2(x)$	=	$\sec^2(x)$	Pythagorean
Pythagorean (cot / csc)	$1 + \cot^2(x)$	=	$\csc^2(x)$	Pythagorean
Cosecant as reciprocal	$\csc(x)$	=	$\dfrac{1}{\sin(x)}$	Reciprocal & quotient
Secant as reciprocal	$\sec(x)$	=	$\dfrac{1}{\cos(x)}$	Reciprocal & quotient
Cotangent as reciprocal of tangent	$\cot(x)$	=	$\dfrac{1}{\tan(x)}$	Reciprocal & quotient
Tangent as quotient	$\tan(x)$	=	$\dfrac{\sin(x)}{\cos(x)}$	Reciprocal & quotient
Cotangent as quotient	$\cot(x)$	=	$\dfrac{\cos(x)}{\sin(x)}$	Reciprocal & quotient
Sine is odd	$\sin(-x)$	=	$-\sin(x)$	Negative-angle
Cosine is even	$\cos(-x)$	=	$\cos(x)$	Negative-angle
Tangent is odd	$\tan(-x)$	=	$-\tan(x)$	Negative-angle
Cotangent is odd	$\cot(-x)$	=	$-\cot(x)$	Negative-angle
Secant is even	$\sec(-x)$	=	$\sec(x)$	Negative-angle
Cosecant is odd	$\csc(-x)$	=	$-\csc(x)$	Negative-angle
Sine of complement = cosine	$\sin\!\left(\tfrac{\pi}{2} - x\right)$	=	$\cos(x)$	Complement (cofunction)
Cosine of complement = sine	$\cos\!\left(\tfrac{\pi}{2} - x\right)$	=	$\sin(x)$	Complement (cofunction)
Tangent of complement = cotangent	$\tan\!\left(\tfrac{\pi}{2} - x\right)$	=	$\cot(x)$	Complement (cofunction)
Cotangent of complement = tangent	$\cot\!\left(\tfrac{\pi}{2} - x\right)$	=	$\tan(x)$	Complement (cofunction)
Secant of complement = cosecant	$\sec\!\left(\tfrac{\pi}{2} - x\right)$	=	$\csc(x)$	Complement (cofunction)
Cosecant of complement = secant	$\csc\!\left(\tfrac{\pi}{2} - x\right)$	=	$\sec(x)$	Complement (cofunction)
Sine of supplement	$\sin(\pi - x)$	=	$\sin(x)$	Supplement
Cosine of supplement	$\cos(\pi - x)$	=	$-\cos(x)$	Supplement
Tangent of supplement	$\tan(\pi - x)$	=	$-\tan(x)$	Supplement
Cotangent of supplement	$\cot(\pi - x)$	=	$-\cot(x)$	Supplement
Secant of supplement	$\sec(\pi - x)$	=	$-\sec(x)$	Supplement
Cosecant of supplement	$\csc(\pi - x)$	=	$\csc(x)$	Supplement
Sine of π/2 + x	$\sin\!\left(\tfrac{\pi}{2} + x\right)$	=	$\cos(x)$	Reference-angle reduction
Cosine of π/2 + x	$\cos\!\left(\tfrac{\pi}{2} + x\right)$	=	$-\sin(x)$	Reference-angle reduction
Tangent of π/2 + x	$\tan\!\left(\tfrac{\pi}{2} + x\right)$	=	$-\cot(x)$	Reference-angle reduction
Cotangent of π/2 + x	$\cot\!\left(\tfrac{\pi}{2} + x\right)$	=	$-\tan(x)$	Reference-angle reduction
Secant of π/2 + x	$\sec\!\left(\tfrac{\pi}{2} + x\right)$	=	$-\csc(x)$	Reference-angle reduction
Cosecant of π/2 + x	$\csc\!\left(\tfrac{\pi}{2} + x\right)$	=	$\sec(x)$	Reference-angle reduction
Sine of π + x	$\sin(\pi + x)$	=	$-\sin(x)$	Reference-angle reduction
Cosine of π + x	$\cos(\pi + x)$	=	$-\cos(x)$	Reference-angle reduction
Tangent of π + x	$\tan(\pi + x)$	=	$\tan(x)$	Reference-angle reduction
Cotangent of π + x	$\cot(\pi + x)$	=	$\cot(x)$	Reference-angle reduction
Secant of π + x	$\sec(\pi + x)$	=	$-\sec(x)$	Reference-angle reduction
Cosecant of π + x	$\csc(\pi + x)$	=	$-\csc(x)$	Reference-angle reduction
Sine of 3π/2 − x	$\sin\!\left(\tfrac{3\pi}{2} - x\right)$	=	$-\cos(x)$	Reference-angle reduction
Cosine of 3π/2 − x	$\cos\!\left(\tfrac{3\pi}{2} - x\right)$	=	$-\sin(x)$	Reference-angle reduction
Tangent of 3π/2 − x	$\tan\!\left(\tfrac{3\pi}{2} - x\right)$	=	$\cot(x)$	Reference-angle reduction
Cotangent of 3π/2 − x	$\cot\!\left(\tfrac{3\pi}{2} - x\right)$	=	$\tan(x)$	Reference-angle reduction
Secant of 3π/2 − x	$\sec\!\left(\tfrac{3\pi}{2} - x\right)$	=	$-\csc(x)$	Reference-angle reduction
Cosecant of 3π/2 − x	$\csc\!\left(\tfrac{3\pi}{2} - x\right)$	=	$-\sec(x)$	Reference-angle reduction
Sine of 3π/2 + x	$\sin\!\left(\tfrac{3\pi}{2} + x\right)$	=	$-\cos(x)$	Reference-angle reduction
Cosine of 3π/2 + x	$\cos\!\left(\tfrac{3\pi}{2} + x\right)$	=	$\sin(x)$	Reference-angle reduction
Tangent of 3π/2 + x	$\tan\!\left(\tfrac{3\pi}{2} + x\right)$	=	$-\cot(x)$	Reference-angle reduction
Cotangent of 3π/2 + x	$\cot\!\left(\tfrac{3\pi}{2} + x\right)$	=	$-\tan(x)$	Reference-angle reduction
Secant of 3π/2 + x	$\sec\!\left(\tfrac{3\pi}{2} + x\right)$	=	$\csc(x)$	Reference-angle reduction
Cosecant of 3π/2 + x	$\csc\!\left(\tfrac{3\pi}{2} + x\right)$	=	$-\sec(x)$	Reference-angle reduction
Sine of 2π − x	$\sin(2\pi - x)$	=	$-\sin(x)$	Reference-angle reduction
Cosine of 2π − x	$\cos(2\pi - x)$	=	$\cos(x)$	Reference-angle reduction
Tangent of 2π − x	$\tan(2\pi - x)$	=	$-\tan(x)$	Reference-angle reduction
Cotangent of 2π − x	$\cot(2\pi - x)$	=	$-\cot(x)$	Reference-angle reduction
Secant of 2π − x	$\sec(2\pi - x)$	=	$\sec(x)$	Reference-angle reduction
Cosecant of 2π − x	$\csc(2\pi - x)$	=	$-\csc(x)$	Reference-angle reduction
Sine of 2π + x	$\sin(2\pi + x)$	=	$\sin(x)$	Reference-angle reduction
Cosine of 2π + x	$\cos(2\pi + x)$	=	$\cos(x)$	Reference-angle reduction
Tangent of 2π + x	$\tan(2\pi + x)$	=	$\tan(x)$	Reference-angle reduction
Cotangent of 2π + x	$\cot(2\pi + x)$	=	$\cot(x)$	Reference-angle reduction
Secant of 2π + x	$\sec(2\pi + x)$	=	$\sec(x)$	Reference-angle reduction
Cosecant of 2π + x	$\csc(2\pi + x)$	=	$\csc(x)$	Reference-angle reduction
Sine of a sum	$\sin(a + b)$	=	$\sin(a)\cos(b) + \cos(a)\sin(b)$	Sum-angle
Cosine of a sum	$\cos(a + b)$	=	$\cos(a)\cos(b) - \sin(a)\sin(b)$	Sum-angle
Tangent of a sum	$\tan(a + b)$	=	$\dfrac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}$	Sum-angle
Cotangent of a sum	$\cot(a + b)$	=	$\dfrac{\cot(a)\cot(b) - 1}{\cot(a) + \cot(b)}$	Sum-angle
Secant of a sum	$\sec(a + b)$	=	$\dfrac{\sec(a)\sec(b)}{1 - \tan(a)\tan(b)}$	Sum-angle
Cosecant of a sum	$\csc(a + b)$	=	$\dfrac{\csc(a)\csc(b)}{\cot(a) + \cot(b)}$	Sum-angle
Sine of a difference	$\sin(a - b)$	=	$\sin(a)\cos(b) - \cos(a)\sin(b)$	Difference-angle
Cosine of a difference	$\cos(a - b)$	=	$\cos(a)\cos(b) + \sin(a)\sin(b)$	Difference-angle
Tangent of a difference	$\tan(a - b)$	=	$\dfrac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}$	Difference-angle
Cotangent of a difference	$\cot(a - b)$	=	$\dfrac{\cot(a)\cot(b) + 1}{\cot(b) - \cot(a)}$	Difference-angle
Secant of a difference	$\sec(a - b)$	=	$\dfrac{\sec(a)\sec(b)}{1 + \tan(a)\tan(b)}$	Difference-angle
Cosecant of a difference	$\csc(a - b)$	=	$\dfrac{\csc(a)\csc(b)}{\cot(b) - \cot(a)}$	Difference-angle
Sine double-angle	$\sin(2x)$	=	$2\sin(x)\cos(x)$	Double-angle
Cosine double-angle	$\cos(2x)$	=	$\cos^2(x) - \sin^2(x)$	Double-angle
Tangent double-angle	$\tan(2x)$	=	$\dfrac{2\tan(x)}{1 - \tan^2(x)}$	Double-angle
Cotangent double-angle	$\cot(2x)$	=	$\dfrac{\cot^2(x) - 1}{2\cot(x)}$	Double-angle
Secant double-angle	$\sec(2x)$	=	$\dfrac{\sec^2(x)}{2 - \sec^2(x)}$	Double-angle
Cosecant double-angle	$\csc(2x)$	=	$\dfrac{\sec(x)\csc(x)}{2}$	Double-angle
Sine triple-angle	$\sin(3x)$	=	$3\sin(x) - 4\sin^3(x)$	Triple-angle
Cosine triple-angle	$\cos(3x)$	=	$4\cos^3(x) - 3\cos(x)$	Triple-angle
Tangent triple-angle	$\tan(3x)$	=	$\dfrac{3\tan(x) - \tan^3(x)}{1 - 3\tan^2(x)}$	Triple-angle
Cotangent triple-angle	$\cot(3x)$	=	$\dfrac{\cot^3(x) - 3\cot(x)}{3\cot^2(x) - 1}$	Triple-angle
Secant triple-angle	$\sec(3x)$	=	$\dfrac{\sec(x)}{4\cos^2(x) - 3}$	Triple-angle
Cosecant triple-angle	$\csc(3x)$	=	$\dfrac{\csc(x)}{3 - 4\sin^2(x)}$	Triple-angle
Sine half-angle	$\sin\!\left(\tfrac{x}{2}\right)$	=	$\pm\sqrt{\dfrac{1 - \cos(x)}{2}}$	Half-angle
Cosine half-angle	$\cos\!\left(\tfrac{x}{2}\right)$	=	$\pm\sqrt{\dfrac{1 + \cos(x)}{2}}$	Half-angle
Tangent half-angle	$\tan\!\left(\tfrac{x}{2}\right)$	=	$\dfrac{1 - \cos(x)}{\sin(x)}$	Half-angle
Cotangent half-angle	$\cot\!\left(\tfrac{x}{2}\right)$	=	$\dfrac{1 + \cos(x)}{\sin(x)}$	Half-angle
Secant half-angle	$\sec\!\left(\tfrac{x}{2}\right)$	=	$\pm\sqrt{\dfrac{2}{1 + \cos(x)}}$	Half-angle
Cosecant half-angle	$\csc\!\left(\tfrac{x}{2}\right)$	=	$\pm\sqrt{\dfrac{2}{1 - \cos(x)}}$	Half-angle
Sine squared	$\sin^2(x)$	=	$\dfrac{1 - \cos(2x)}{2}$	Power reduction
Cosine squared	$\cos^2(x)$	=	$\dfrac{1 + \cos(2x)}{2}$	Power reduction
Tangent squared	$\tan^2(x)$	=	$\dfrac{1 - \cos(2x)}{1 + \cos(2x)}$	Power reduction
Sine times cosine	$\sin(a)\cos(b)$	=	$\tfrac{1}{2}\bigl[\sin(a + b) + \sin(a - b)\bigr]$	Product-to-sum
Cosine times sine	$\cos(a)\sin(b)$	=	$\tfrac{1}{2}\bigl[\sin(a + b) - \sin(a - b)\bigr]$	Product-to-sum
Cosine times cosine	$\cos(a)\cos(b)$	=	$\tfrac{1}{2}\bigl[\cos(a - b) + \cos(a + b)\bigr]$	Product-to-sum
Sine times sine	$\sin(a)\sin(b)$	=	$\tfrac{1}{2}\bigl[\cos(a - b) - \cos(a + b)\bigr]$	Product-to-sum
Sine plus sine	$\sin(a) + \sin(b)$	=	$2\sin\!\left(\tfrac{a + b}{2}\right)\cos\!\left(\tfrac{a - b}{2}\right)$	Sum-to-product
Sine minus sine	$\sin(a) - \sin(b)$	=	$2\cos\!\left(\tfrac{a + b}{2}\right)\sin\!\left(\tfrac{a - b}{2}\right)$	Sum-to-product
Cosine plus cosine	$\cos(a) + \cos(b)$	=	$2\cos\!\left(\tfrac{a + b}{2}\right)\cos\!\left(\tfrac{a - b}{2}\right)$	Sum-to-product
Cosine minus cosine	$\cos(a) - \cos(b)$	=	$-2\sin\!\left(\tfrac{a + b}{2}\right)\sin\!\left(\tfrac{a - b}{2}\right)$	Sum-to-product
Sine of arcsine	$\sin(\arcsin x)$	=	$x$	Inverse
Cosine of arccosine	$\cos(\arccos x)$	=	$x$	Inverse
Tangent of arctangent	$\tan(\arctan x)$	=	$x$	Inverse
Cosine of arcsine	$\cos(\arcsin x)$	=	$\sqrt{1 - x^2}$	Inverse
Sine of arccosine	$\sin(\arccos x)$	=	$\sqrt{1 - x^2}$	Inverse
Tangent of arcsine	$\tan(\arcsin x)$	=	$\dfrac{x}{\sqrt{1 - x^2}}$	Inverse
Arcsine plus arccosine	$\arcsin(x) + \arccos(x)$	=	$\tfrac{\pi}{2}$	Inverse
Arctangent plus arccotangent	$\arctan(x) + \operatorname{arccot}(x)$	=	$\tfrac{\pi}{2}$	Inverse
Arcsecant plus arccosecant	$\operatorname{arcsec}(x) + \operatorname{arccsc}(x)$	=	$\tfrac{\pi}{2}$	Inverse
Arcsine of negative	$\arcsin(-x)$	=	$-\arcsin(x)$	Inverse
Arccosine of negative	$\arccos(-x)$	=	$\pi - \arccos(x)$	Inverse
Arctangent of negative	$\arctan(-x)$	=	$-\arctan(x)$	Inverse

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 $\cos$ .

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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.

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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.

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Double-angle

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

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Triple-angle

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

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x/2

Half-angle

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

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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: $\sin$ , $\tan$ , $\cot$ , $\csc$ .

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even

Even-function identities

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

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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}