Trigonometric identities are equations involving trigonometric functions that are true for all values in their domains.
Knowing those identities and understanding them is important because they:
▪ Simplify complex expressions
▪ Help solving equations not easily solvable in their original form
▪ May be useful in proving mathematical theorems
▪ Model periodic phenomena in physics, engineering, and other fields
▪ Transform expressions to more useful forms for integration or differentiation

Definition-based Identities

Definition-based identities derive from the fundamental relationships between trigonometric functions based on their right-triangle definitions.
Those may be well illustrated by unit circle.

law	formula	explanation
Definition of Sine	sin(θ) = opposite / hypotenuse	Sine defined from right triangle
Definition of Cosine	cos(θ) = adjacent / hypotenuse	Cosine defined from right triangle
Definition of Tangent	tan(θ) = opposite / adjacent	Tangent defined from right triangle
Definition of Cotangent	cot(θ) = adjacent / opposite	Reciprocal of tangent
Definition of Secant	sec(θ) = hypotenuse / adjacent	Reciprocal of cosine
Definition of Cosecant	csc(θ) = hypotenuse / opposite	Reciprocal of sine
Tangent-Sine-Cosine Relation	tan(θ) = sin(θ) / cos(θ)	Derived from basic definitions
Cotangent-Sine-Cosine Relation	cot(θ) = cos(θ) / sin(θ)	Derived from basic definitions

Definition-based identities are fundamental relationships derived directly from how trigonometric functions are defined in a right triangle. They establish the basic connections between these functions and serve as building blocks for more complex identities.
To get better feeling and understanding of basic definition-based identities use our visual intteractive unit circle tool.

Reciprocal Identities

In mathematics, the reciprocal of a number or expression is 1 divided by that number.
So, for any non-zero value

𝑥

, the reciprocal is

\frac{1}{x}

.
A reciprocal identity expresses the relationship between a trigonometric function and its multiplicative inverse — basically, how each function "flips".
Each basic trigonometric function (sine, cosine, tangent) has a reciprocal counterpart.

law	formula	explanation
Reciprocal of Sine	sin(θ) = 1 / csc(θ)	Sine is reciprocal of cosecant
Reciprocal of Cosine	cos(θ) = 1 / sec(θ)	Cosine is reciprocal of secant
Reciprocal of Tangent	tan(θ) = 1 / cot(θ)	Tangent is reciprocal of cotangent
Reciprocal of Cosecant	csc(θ) = 1 / sin(θ)	Cosecant is reciprocal of sine
Reciprocal of Secant	sec(θ) = 1 / cos(θ)	Secant is reciprocal of cosine
Reciprocal of Cotangent	cot(θ) = 1 / tan(θ)	Cotangent is reciprocal of tangent

Reciprocal identities help simplify rational trigonometric expressions, especially in algebra-heavy problems or when combining functions.
Knowing how to flip between functions is critical in isolating variables and solving identities.
Reciprocal identities let you flip between familiar and less familiar functions, extend your solving toolbox, and reduce redundancy in learning.

Pythagorean Identities

Pythagorean Identities are based on the Pythagorean theorem applied to the unit circle.
They express fundamental relationships that always equal 1 or have the form "1 + something²". The core identity sin²(θ) + cos²(θ) = 1 comes directly from the unit circle where any point (cos θ, sin θ) satisfies x² + y² = 1. The other two identities are derived by dividing this fundamental relationship by cos² or sin².

law	formula	explanation
Sine and Cosine	sin²(θ) + cos²(θ) = 1	Fundamental identity derived directly from the unit circle .
Tangent and Secant	1 + tan²(θ) = sec²(θ)	Derived from dividing sin² + cos² = 1 by cos²
Cotangent and Cosecant	1 + cot²(θ) = csc²(θ)	Derived from dividing sin² + cos² = 1 by sin²

Being familiar with these identities is a crucial skill because they allow converting between different trigonometric functions and are essential for simplifying complex trigonometric expressions.

Triangle Geometry Identities

Triangle Geometry Identities are fundamental relationships that apply to any triangle, not just right triangles. They're special because they connect angles and side lengths in general triangles, extending beyond basic trigonometry. The Law of Sines relates ratios of sides to opposite angles, while the Law of Cosines generalizes the Pythagorean theorem for non-right triangles.

law	formula	explanation
Law of Sines	sin(A)/a = sin(B)/b = sin(C)/c	Relates the ratios of angles and opposite sides in any triangle
Law of Cosines (Standard Form)	c² = a² + b² − 2ab·cos(C)	Generalization of the Pythagorean Theorem for any triangle
Law of Cosines (Alternative Forms)	a² = b² + c² − 2bc·cos(A); b² = a² + c² − 2ac·cos(B)	Same law applied for other sides of the triangle

These identities are crucial for solving real-world problems involving triangulation, navigation, and engineering where you need to find unknown sides or angles in any triangle configuration.

Even-Odd Identities

Even-Odd Identities describe the symmetry properties of trigonometric functions when the angle is negated. They're special because they classify functions as either even (symmetric about the y-axis) or odd (symmetric about the origin). Cosine and secant are even functions, meaning f(-θ) = f(θ), while sine, tangent, cosecant, and cotangent are odd functions, meaning f(-θ) = -f(θ).

law	formula	explanation
Odd Sine	sin(−θ) = −sin(θ)	Sine is an odd function; reflects through origin
Even Cosine	cos(−θ) = cos(θ)	Cosine is an even function; symmetric about y-axis
Odd Tangent	tan(−θ) = −tan(θ)	Tangent is odd; sine is odd and cosine is even
Odd Cosecant	csc(−θ) = −csc(θ)	Cosecant is reciprocal of sine, so also odd
Even Secant	sec(−θ) = sec(θ)	Secant is reciprocal of cosine, so also even
Odd Cotangent	cot(−θ) = −cot(θ)	Cotangent is reciprocal of tangent, so also odd

These identities are important for simplifying expressions with negative angles, understanding function behavior, and solving equations where angle direction matters.

Co-Function Identities

Co-Function Identities express the relationship between trigonometric functions and their "co-functions" (complementary functions). They're special because they show how trig functions of complementary angles (angles that add to 90°) are related. For example, sine of an angle equals cosine of its complement. These identities exist because in a right triangle, the two acute angles are complementary, so one angle's opposite side becomes the other angle's adjacent side.

law	formula	explanation
Sine of Complement	sin(π/2 − θ) = cos(θ)	Sine of an angle's complement equals cosine
Cosine of Complement	cos(π/2 − θ) = sin(θ)	Cosine of an angle's complement equals sine
Tangent of Complement	tan(π/2 − θ) = cot(θ)	Tangent of complement equals cotangent
Cotangent of Complement	cot(π/2 − θ) = tan(θ)	Cotangent of complement equals tangent
Secant of Complement	sec(π/2 − θ) = csc(θ)	Secant of complement equals cosecant
Cosecant of Complement	csc(π/2 − θ) = sec(θ)	Cosecant of complement equals secant

Co-Function identities are important for converting between different trig functions and simplifying expressions involving complementary angles.

Periodicity Identities

Periodicity Identities describe how trigonometric functions repeat their values at regular intervals. They're special because they capture the cyclical nature of trigonometric functions - sine and cosine repeat every 2π (full circle), while tangent and cotangent repeat every π (half circle). These identities exist because trigonometric functions are based on circular motion and angles, which naturally cycle.

law	formula	explanation
Sine Periodicity	sin(θ + 2π) = sin(θ)	Sine repeats every 2π
Cosine Periodicity	cos(θ + 2π) = cos(θ)	Cosine repeats every 2π
Tangent Periodicity	tan(θ + π) = tan(θ)	Tangent repeats every π
Cotangent Periodicity	cot(θ + π) = cot(θ)	Cotangent repeats every π
Secant Periodicity	sec(θ + 2π) = sec(θ)	Secant repeats every 2π
Cosecant Periodicity	csc(θ + 2π) = csc(θ)	Cosecant repeats every 2π

Understanding periodicity identities is crucial for solving equations with multiple solutions, understanding wave behavior, and working with any angle by reducing it to a standard interval.

Shift Identities

Shift Identities show how trigonometric functions behave when the angle is shifted by specific amounts like π or π/2. They're special because they reveal the phase relationships between different trig functions - for example, how sine becomes cosine when shifted by π/2. These identities exist because of the geometric relationships on the unit circle where shifting an angle corresponds to rotating the point.

law	formula	explanation
Sine Shift by π	sin(θ + π) = −sin(θ)	Sine shifts by π with a sign flip
Cosine Shift by π	cos(θ + π) = −cos(θ)	Cosine shifts by π with a sign flip
Tangent Shift by π	tan(θ + π) = tan(θ)	Tangent is periodic with π and does not flip
Sine Shift by π/2	sin(θ + π/2) = cos(θ)	Sine shifted by π/2 becomes cosine
Cosine Shift by π/2	cos(θ + π/2) = −sin(θ)	Cosine shifted by π/2 becomes negative sine
Tangent Shift by π/2	tan(θ + π/2) = −cot(θ)	Tangent shifted by π/2 becomes negative cotangent

Shift identities are important for understanding wave phase shifts, converting between trig functions, and analyzing periodic phenomena where timing or phase differences matter.

Angle Sum Identities

Angle Sum Identities express trigonometric functions of the sum of two angles in terms of the individual angles' functions. They're special because they break down complex angle combinations into manageable parts using products and sums of simpler functions. These identities exist due to the geometric properties of rotating and combining angles on the unit circle.

law	formula	explanation
Sine of Sum	sin(a + b) = sin(a)cos(b) + cos(a)sin(b)	Expands sine of a sum
Cosine of Sum	cos(a + b) = cos(a)cos(b) − sin(a)sin(b)	Expands cosine of a sum
Tangent of Sum	tan(a + b) = (tan(a) + tan(b)) / (1 − tan(a)tan(b))	Expands tangent of a sum
Cotangent of Sum	cot(a + b) = (cot(a)cot(b) − 1) / (cot(a) + cot(b))	Expands cotangent of a sum
Secant of Sum	sec(a + b) = 1 / [cos(a + b)]	Reciprocal of cosine of sum
Cosecant of Sum	csc(a + b) = 1 / [sin(a + b)]	Reciprocal of sine of sum

These identities are fundamental for calculus, solving trigonometric equations, proving other identities, and applications in physics where multiple rotations or oscillations combine, such as in wave interference and vector analysis.

Angle Difference Identities

Angle Difference Identities express trigonometric functions of the difference between two angles in terms of the individual angles' functions. They're special because they complement the sum identities, providing the complete toolkit for breaking down angle combinations. These identities exist from the same geometric principles as sum identities but with opposite rotation directions on the unit circle.

law	formula	explanation
Sine of Difference	sin(a − b) = sin(a)cos(b) − cos(a)sin(b)	Expands sine of a difference
Cosine of Difference	cos(a − b) = cos(a)cos(b) + sin(a)sin(b)	Expands cosine of a difference
Tangent of Difference	tan(a − b) = (tan(a) − tan(b)) / (1 + tan(a)tan(b))	Expands tangent of a difference
Cotangent of Difference	cot(a − b) = (cot(a)cot(b) + 1) / (cot(b) − cot(a))	Expands cotangent of a difference
Secant of Difference	sec(a − b) = 1 / [cos(a − b)]	Reciprocal of cosine of difference
Cosecant of Difference	csc(a − b) = 1 / [sin(a − b)]	Reciprocal of sine of difference

Angle Difference identities are essential for solving trigonometric equations, deriving other identities, and applications involving relative motion, phase differences, or when one oscillation opposes another in physics and engineering problems.

Double Angle Identities

Double Angle Identities are specialized cases of angle sum formulas where both angles are identical (θ + θ = 2θ). What makes them unique is their simplified, elegant forms that directly relate single-angle functions to double-angle functions. They emerge naturally from setting a = b in the sum identities, creating powerful shortcuts.

law	formula	explanation
Double Angle for Sine	sin(2θ) = 2sin(θ)cos(θ)	Derived from the sine of a sum
Double Angle for Cosine	cos(2θ) = cos²(θ) − sin²(θ)	Standard form from the cosine of a sum
Double Angle for Cosine	cos(2θ) = 2cos²(θ) − 1	Alternate form using cosine only
Double Angle for Cosine	cos(2θ) = 1 − 2sin²(θ)	Alternate form using sine only
Double Angle for Tangent	tan(2θ) = (2tan(θ)) / (1 − tan²(θ))	Derived from the tangent of a sum
Double Angle for Cotangent	cot(2θ) = (cot²(θ) − 1) / (2cot(θ))	Derived from reciprocal of tangent double angle

These identities are vital for integration in calculus, solving equations with multiple angle relationships, and analyzing phenomena with frequency doubling, such as harmonics in music, optical frequency conversion, and engineering systems with gear ratios.

Half Angle Identities

Half Angle Identities work in reverse of double angle formulas, expressing functions of θ/2 in terms of functions of θ. Their uniqueness lies in providing square root forms and rational expressions that eliminate half-angles from equations. They arise by algebraically rearranging double angle identities and often involve ± signs requiring careful quadrant analysis.

law	formula	explanation
Half Angle for Sine	sin²(θ) = (1 − cos(2θ)) / 2	Derived from cosine double angle identity
Half Angle for Cosine	cos²(θ) = (1 + cos(2θ)) / 2	Derived from cosine double angle identity
Half Angle for Tangent	tan(θ/2) = ±√[(1 − cos(θ)) / (1 + cos(θ))]	Square root form derived from sine and cosine
Half Angle for Tangent	tan(θ/2) = sin(θ) / (1 + cos(θ))	Derived by rationalizing tangent expression
Half Angle for Tangent	tan(θ/2) = (1 − cos(θ)) / sin(θ)	Alternate rational form for tangent
Half Angle for Cotangent	cot(θ/2) = ±√[(1 + cos(θ)) / (1 − cos(θ))]	Reciprocal of the square root tangent identity

These identities are indispensable for integration techniques, solving equations with fractional angles, and applications involving bisection problems, such as cutting angles in half for construction, optics with half-wave plates, and signal processing with subharmonics.

Triple Angle Identities

Triple Angle Identities extend angle multiplication to express functions of 3θ in terms of functions of θ. They're distinctive for producing cubic polynomial relationships, creating elegant algebraic forms like sin(3θ) = 3sin(θ) - 4sin³(θ). These identities emerge from repeatedly applying angle sum formulas or using complex number methods.

law	formula	explanation
Triple Angle for Sine	sin(3θ) = 3sin(θ) − 4sin³(θ)	Expanded using angle addition identity
Triple Angle for Cosine	cos(3θ) = 4cos³(θ) − 3cos(θ)	Expanded using angle addition identity
Triple Angle for Tangent	tan(3θ) = (3tan(θ) − tan³(θ)) / (1 − 3tan²(θ))	Derived from tangent sum identity
Triple Angle for Cotangent	cot(3θ) = (cot³(θ) − 3cot(θ)) / (3cot²(θ) − 1)	Derived from reciprocal of tangent triple angle
Triple Angle for Secant	sec(3θ) = 1 / (4cos³(θ) − 3cos(θ))	Reciprocal of cosine triple angle
Triple Angle for Cosecant	csc(3θ) = 1 / (3sin(θ) − 4sin³(θ))	Reciprocal of sine triple angle

Those identities are valuable in advanced calculus, solving higher-order trigonometric equations, and specialized applications like three-phase electrical systems, crystallography with threefold symmetry, and acoustics where third harmonics play crucial roles in sound quality and instrument timbre.

Power-Reducing Identities

Power-Reducing Identities convert higher powers of trigonometric functions into expressions with lower powers and multiple angles. They're special because they eliminate troublesome squared and higher-power terms by expressing them using first-power functions of double angles.

law	formula	explanation
Power-Reducing for Sine	sin²(θ) = (1 − cos(2θ)) / 2	Reduces power of sine using double angle
Power-Reducing for Cosine	cos²(θ) = (1 + cos(2θ)) / 2	Reduces power of cosine using double angle
Power-Reducing for Tangent	tan²(θ) = (1 − cos(2θ)) / (1 + cos(2θ))	Derived from sin²/cos² power-reduced forms
Power-Reducing for Cotangent	cot²(θ) = (1 + cos(2θ)) / (1 − cos(2θ))	Reciprocal form of tangent power-reduction
Power-Reducing for Secant	sec²(θ) = 2 / (1 + cos(2θ))	Derived by inverting cos² power identity
Power-Reducing for Cosecant	csc²(θ) = 2 / (1 − cos(2θ))	Derived by inverting sin² power identity

These identities stem from rearranging double angle formulas to isolate the squared terms. They're essential in calculus integration where powers create complexity, Fourier analysis for breaking down periodic functions, and signal processing where reducing power terms simplifies frequency domain analysis and filtering operations.

Product-to-Sum Identities

Product-to-Sum Identities transform products of trigonometric functions into sums or differences of simpler trigonometric expressions. They're special because they convert multiplication (which is complex) into addition (which is simpler), essentially "unpacking" products into linear combinations. These identities arise from algebraically manipulating angle sum and difference formulas.

law	formula	explanation
Product-to-Sum for sin(a)sin(b)	sin(a)sin(b) = ½[cos(a − b) − cos(a + b)]	Converts product of sines into sum of cosines
Product-to-Sum for cos(a)cos(b)	cos(a)cos(b) = ½[cos(a − b) + cos(a + b)]	Converts product of cosines into sum of cosines
Product-to-Sum for sin(a)cos(b)	sin(a)cos(b) = ½[sin(a + b) + sin(a − b)]	Converts product of sine and cosine into sum of sines
Product-to-Sum for cos(a)sin(b)	cos(a)sin(b) = ½[sin(a + b) − sin(a − b)]	Converts product of cosine and sine into difference of sines
Product-to-Sum for tan(a)tan(b)	tan(a)tan(b) = [cos(a − b) − cos(a + b)] / [cos(a − b) + cos(a + b)]	Derived using tangent in terms of sine and cosine
Product-to-Sum for cot(a)cot(b)	cot(a)cot(b) = [cos(a − b) + cos(a + b)] / [cos(a − b) − cos(a + b)]	Derived using cotangent in terms of cosine and sine

These identities are crucial in signal processing for analyzing beat frequencies, Fourier analysis for decomposing complex waveforms, acoustics for understanding interference patterns, and integration techniques where products are harder to handle than sums.

Sum-to-Product Identities

Sum-to-Product Identities convert sums and differences of trigonometric functions into products of simpler expressions. They're special because they perform the reverse operation of product-to-sum identities, transforming linear combinations into multiplicative forms. These identities emerge from reversing the algebraic manipulation of angle sum formulas. They're valuable for factoring trigonometric expressions, solving equations where products reveal common factors, analyzing wave interference where constructive and destructive patterns create amplitude modulation, and in acoustics for understanding how combined frequencies create beats and harmonics.

law	formula	explanation
Sum-to-Product for sin(a) + sin(b)	sin(a) + sin(b) = 2sin[(a + b)/2]cos[(a − b)/2]	Converts sum of sines into product of sine and cosine
Sum-to-Product for sin(a) − sin(b)	sin(a) − sin(b) = 2cos[(a + b)/2]sin[(a − b)/2]	Converts difference of sines into product
Sum-to-Product for cos(a) + cos(b)	cos(a) + cos(b) = 2cos[(a + b)/2]cos[(a − b)/2]	Converts sum of cosines into product
Sum-to-Product for cos(a) − cos(b)	cos(a) − cos(b) = −2sin[(a + b)/2]sin[(a − b)/2]	Converts difference of cosines into negative product of sines
Sum-to-Product for tan(a) + tan(b)	tan(a) + tan(b) = sin(a + b) / [cos(a)cos(b)]	Derived by expressing tangent in terms of sine and cosine
Sum-to-Product for tan(a) − tan(b)	tan(a) − tan(b) = sin(a − b) / [cos(a)cos(b)]	Difference of tangents expressed as sine over product of cosines