Trigonometric Identities — Formulas, Proofs, Examples

Trigonometry

TL;DR

Trigonometric identities are equations involving sine, cosine, tangent and their reciprocals that hold for every angle in their domain — the algebraic glue between the six trig functions. This article covers the eight identity families (reciprocal, quotient, Pythagorean, co-function, even-odd, sum-difference, double-angle, half-angle, product-to-sum), the unit-circle proof behind each, three worked examples in both degrees and radians, and the sign-flip mistakes that cost the most marks.

BT

Bhanzu Team Last updated on June 1, 202616 min read

One Equation Hidden Inside Every Wave You Have Ever Heard

A single identity — sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 is behind GPS, Wi-Fi, MRI scans, and every musical chord on a guitar.

Trigonometric identities are equations involving the six trigonometric functions — sine, cosine, tangent, cosecant, secant, cotangent — that hold true for every angle in their domain. Where an equation like sin⁡θ=1/2 is true only at specific angles, an identity like sin⁡2θ+cos⁡2θ=1 is true at every angle. That distinction is the whole reason identities matter: they are the rewrite rules that let you simplify, prove, and solve every other trig problem you will meet.

The full library breaks into eight families. They link the six functions to each other, to angle sums and differences, to doubled and halved angles, and to products of trig values.

The Eight Identity Families at a Glance

Quick facts.

• Eight families — reciprocal, quotient, Pythagorean, co-function, even-odd, sum-difference, double-angle, half-angle (with product-to-sum and sum-to-product as a derived ninth pair).
• Three foundational Pythagorean identities — derived directly from x2+y2=1 on the unit circle.
• All identities hold for any real angle in the domain — degree or radian measure works identically; the geometry is the same.
• Domain caveats matter — tan⁡θ and sec⁡θ are undefined at θ=π/2+nπ; cot⁡θ and csc⁡θ are undefined at θ=nπ.
• Grade introduced: CCSS-M F-TF.C.8 and F-TF.C.9 (Pythagorean, sum-difference, double-angle identities); NCERT Class 11 Chapter 3 — Trigonometric Functions (full identity catalogue).

Family 1 — Reciprocal identities

The three reciprocal pairs reduce six functions to two.

csc⁡θ=1/sin⁡θ, sec⁡θ=1/cos⁡θ, cot⁡θ=1/tan⁡θ.
A co-function ( csc, sec, cot) is always the reciprocal of its non-co partner. The matched pairs are sine with cosecant, cosine with secant, tangent with cotangent. The "co" prefix tags the reciprocal, not the complementary angle.

Family 2 — Quotient identities

The tangent and cotangent are not new functions in any meaningful sense — they are ratios of sine and cosine.

tan⁡θ=sin⁡θ/cos⁡θ, cot⁡θ=cos⁡θ/sin⁡θ.
Every proof you ever write that involves tangent eventually rewrites it as sin⁡/cos⁡. The quotient identity is the bridge.

Family 3 — Pythagorean identities

Three identities follow from the unit-circle equation x2+y2=1. Setting x=cos⁡θ and y=sin⁡θ gives the first; dividing it by cos⁡2θ gives the second; dividing by sin⁡2θ gives the third.

;sin⁡2θ+cos⁡2θ=1;
;1+tan⁡2θ=sec⁡2θ;
;1+cot⁡2θ=csc⁡2θ;

The third identity is the one students forget most often.

Family 4 — Co-function identities

Every trig function equals its co-function evaluated at the complementary angle.

sin⁡θ=cos⁡(π/2−θ), cos⁡θ=sin⁡(π/2−θ),
tan⁡θ=cot⁡(π/2−θ), cot⁡θ=tan⁡(π/2−θ),
sec⁡θ=csc⁡(π/2−θ), csc⁡θ=sec⁡(π/2−θ).

In degrees, replace π/2 with 90°.

Family 5 — Even-odd (negative-angle) identities

Cosine and secant are even — symmetric across the y-axis on the unit circle. The other four are odd — they flip sign when the angle does.

cos⁡(−θ)=cos⁡θ, sec⁡(−θ)=sec⁡θ,
sin⁡(−θ)=−sin⁡θ, tan⁡(−θ)=−tan⁡θ,
csc⁡(−θ)=−csc⁡θ, cot⁡(−θ)=−cot⁡θ.

Family 6 — Sum and difference identities

Six identities that compute trig of A±B from trig of A and B separately.

sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B, cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B, tan⁡(A±B)=tan⁡A±tan⁡B/(1∓tan⁡Atan⁡B).

Family 7 — Double-angle identities

Set A=B=θ in the sum identities and you get:

sin⁡(2θ)=2sin⁡θcos⁡θ, cos⁡(2θ)=cos⁡2θ−sin⁡2θ=1−2sin⁡2θ=2cos⁡2θ−1, tan⁡(2θ)=2tan⁡θ/(1−tan⁡2θ).

Family 8 — Half-angle identities

Rearrange the double-angle identities and you can compute trig of θ/2:

sin⁡(θ/2)=±√(1−cosθ)/2, cos⁡(θ/2)=±√(1+cosθ)/2, tan⁡(θ/2)=(1−cosθ)/sinθ=(sinθ)/(1+cosθ).

Family 9 — Product-to-sum and sum-to-product

The product-to-sum identities turn a product of two trig values into a sum (or difference).

sin⁡Acos⁡B=1/2[sin⁡(A+B)+sin⁡(A−B)], cos⁡Acos⁡B=1/2[cos⁡(A−B)+cos⁡(A+B)], sin⁡Asin⁡B=1/2[cos⁡(A−B)−cos⁡(A+B)].

Why the Pythagorean Identity Falls Out of the Unit Circle

The single most important identity is sin⁡2θ+cos⁡2θ=1, and the proof is one line — provided you start at the right place.

A unit circle is the circle of radius 1 centered at the origin. Any point on it satisfies x2+y2=1. The standard definitions of cosine and sine are the coordinates of that point:

x=cos⁡θ,y=sin⁡θ for the point at angle θ.

Substitute and you get cos⁡2θ+sin⁡2θ=1.

Double-Anchoring — Right Triangle and Unit Circle

Every trigonometric identity has two readings: the right-triangle reading and the unit-circle reading.

From the right triangle.

Three Worked Examples of Trigonometric Identities

Quick. Simplify sin⁡2(30°)+cos⁡2(30°) without computing the values.

By the first Pythagorean identity, sin⁡2θ+cos⁡2θ=1 for every angle θ. So:

sin⁡2(30°)+cos⁡2(30°)=1.

Final answer: 1.

Standard (Wrong Path First — Watch How This Goes Wrong).

Verify the identity 1−cos⁡2θ/sin⁡θ=sin⁡θ for sin⁡θ≠0.

The wrong path. A student tries to manipulate both sides of the equation:

1−cos⁡2θ/sin⁡θ=sin⁡θ.

They multiply both sides by sin⁡θ:

1−cos⁡2θ=sin⁡2θ.

They conclude "that's the Pythagorean identity rearranged, so done." But cross-multiplying across the equals sign during an identity proof is forbidden — that's the equation move, not the identity move.

Final answer: verified — for every θ with sin⁡θ≠0, 1−cos⁡2θ/sin⁡θ=sin⁡θ.

Where Trigonometric Identities Earn Their Living

The identities are not a memorisation drill for an exam; they are the rewrite engine behind nearly every wave-based technology in modern life:

• GPS and triangulation.
• Signal processing — Fourier analysis.
• Electrical engineering — AC power.
• Optics and interference.
• Computer graphics.
• MRI imaging.

The Mathematicians Who Shaped Trigonometric Identities

• Hipparchus of Nicaea
• Madhava of Sangamagrama
• Leonhard Euler

Tripping Points to Watch For

1. Confusing co-function with reciprocal

2. Forgetting the third Pythagorean identity

3. Flipping signs in the cosine sum-and-difference identity

4. Using "both-sides" algebra to prove an identity

Conclusion

• Trigonometric identities are equations between trig functions that hold for every angle in their domain.
• The eight families — reciprocal, quotient, Pythagorean, co-function, even-odd, sum-difference, double-angle, half-angle — all derive from the unit-circle equation x2+y2=1 plus the definitions of the six functions.
• The most marks lost come from cross-multiplying across the equals sign during a proof and from flipping signs in the cosine sum-difference rule.

Take Trigonometric Identities for a Test Drive

Try these three before moving on. Each one targets a different family — work them in both degrees and radians, then check your sign on the cosine identity.