Where does ^2 + ^2 = 1 come from?

From Pythagoras on the unit circle. The point (, ) is a distance 1 from the origin, so ^2 + ^2 = 1. It is Pythagoras in disguise.

Back to course

E4PureCore

Trigonometric Identities

Q: What does ^2 mean?

It means ( )^2 — the sine of the angle, then squared. It is a common slip to misread it as ( ^2).

Two identities do an enormous amount of work in trigonometry: $tan θ = \frac{s i n θ}{c o s θ}$ and $sin^{2} θ + cos^{2} θ = 1$ . They let you simplify expressions, prove results, and rewrite equations so they can actually be solved.

26 min Video by TLMaths Trigonometry

The big picture

An identity is an equation true for $every$ value of the variable — not something to solve, but a tool for rewriting. The two you meet here both fall straight out of the unit circle: the quotient identity from the definition of tan, and the Pythagorean identity from Pythagoras applied to the unit-circle point $(cos θ, sin θ)$ . Their power is transformation. A messy trig expression becomes simple, and an unsolvable-looking equation (mixing $sin$ and $cos$ ) becomes a familiar quadratic once an identity converts everything to a single ratio. Fluency here is what makes the next lesson, solving trig equations, tractable.

A-Level Maths E5-01: Proving tanθ = sinθ / cosθWatch the walkthrough, then work through the notes below.

Open on YouTube

What you'll be able to do

State and derive the quotient and Pythagorean identities
Use identities to simplify trigonometric expressions
Prove trigonometric identities by manipulating one side
Rewrite an equation into a single trig ratio using an identity

The two core identities and where they come from

The $quotient identity$ $tan θ = \frac{s i n θ}{c o s θ}$ follows directly from the unit circle: tan is the ratio of the $y$ - to the $x$ -coordinate.

The $Pythagorean identity$ $sin^{2} θ + cos^{2} θ = 1$ is Pythagoras itself. The unit-circle point $(cos θ, sin θ)$ is distance 1 from the origin, so $cos^{2} θ + sin^{2} θ = 1^{2}$ . (Note $sin^{2} θ$ means $(sin θ)^{2}$ .)

tan θ = \frac{sin θ}{cos θ}

Quotient identity.

sin^{2} θ + cos^{2} θ = 1

Pythagorean identity — from the unit circle.

Why this works

The Pythagorean identity is not a formula to memorise blindly — it is literally Pythagoras on the unit circle. Seeing that makes it (and its rearrangements $sin^{2} θ = 1 - cos^{2} θ$ ) unforgettable.

Simplifying expressions

The commonest use is to reduce a complicated expression. Swap $tan$ for $\frac{s i n}{c o s}$ , or replace $sin^{2} θ$ with $1 - cos^{2} θ$ (or vice versa), and watch terms cancel or combine.

1Replace

tan θ = \frac{s i n θ}{c o s θ}

\frac{sin θ}{sin θ / cos θ}

2Dividing by a fraction:

sin θ \times \frac{cos θ}{sin θ}

3The

sin θ

cancels, leaving

cos θ

Answer

cos θ

Tip — When an expression mixes $sin$ , $cos$ and $tan$ , converting everything to $sin$ and $cos$ first almost always reveals the simplification.

Proving identities

To $prove$ an identity, start from one side (usually the more complicated one) and manipulate it until it matches the other side. Do not treat it like an equation to “solve” by moving terms across the equals sign — that assumes what you are trying to prove.

1Start with the left side and expand:

sin^{2} θ + 2 sin θ cos θ + cos^{2} θ

2Group the squares:

(sin^{2} θ + cos^{2} θ) + 2 sin θ cos θ

3Use the Pythagorean identity:

1 + 2 sin θ cos θ

4This equals the right side.

■

AnswerLHS simplifies to

1 + 2 sin θ cos θ =

RHS.

Why this works

Working one side down to the other is the safe method: it never assumes the result. Spot the $sin^{2} θ + cos^{2} θ$ pattern — collapsing it to 1 is the move that finishes most identity proofs.

Think like an examiner

To prove an identity, manipulate one side to the other; do not rearrange across the equals sign as if solving.
Recognise $sin^{2} θ + cos^{2} θ$ and replace it with 1 — it is the key simplifying step.
Convert $tan$ (and any reciprocal ratios) into $sin$ and $cos$ when stuck.
Remember $sin^{2} θ = (sin θ)^{2}$ , not $sin (θ^{2})$ — a notation trap.

Common misconceptions

Reading

sin^{2} θ

sin (θ^{2})

sin^{2} θ

means

(sin θ)^{2}

— square the whole ratio.

Proving an identity by moving terms across the

=

sign.

Start on one side and transform it into the other; never assume the identity holds.

Writing

sin^{2} θ + cos^{2} θ = θ

or some other value.

It always equals exactly 1.

Forgetting the rearrangements

sin^{2} θ = 1 - cos^{2} θ

Both rearrangements are fair game and often the needed step.

The identities

tan θ = \frac{sin θ}{cos θ}

Quotient identity.

sin^{2} θ + cos^{2} θ = 1

Pythagorean identity.

sin^{2} θ = 1 - cos^{2} θ

A useful rearrangement.

Stretch yourself

Prove that $\frac{1 - cos ^{2} θ}{sin θ cos θ} = tan θ$ .

Hint — Replace $1 - cos^{2} θ$ using the Pythagorean identity, then cancel and use the quotient identity.

Questions students ask

What is a trigonometric identity?

An equation involving trig ratios that is true for every value of the angle. Unlike an equation you solve, an identity is a rewriting rule you use to simplify or prove other statements.

Where does

sin^{2} θ + cos^{2} θ = 1

come from?

From Pythagoras on the unit circle. The point

(cos θ, sin θ)

is a distance 1 from the origin, so

cos^{2} θ + sin^{2} θ = 1

. It is Pythagoras in disguise.

How do I prove an identity?

Take one side — usually the messier one — and manipulate it step by step until it becomes the other side. Do not move terms across the equals sign, as that assumes the very thing you are proving.

What does

sin^{2} θ

mean?

It means

(sin θ)^{2}

— the sine of the angle, then squared. It is a common slip to misread it as

sin (θ^{2})

Key takeaways

Two core identities: $tan θ = \frac{s i n θ}{c o s θ}$ and $sin^{2} θ + cos^{2} θ = 1$ .
Both come straight from the unit circle (tan ratio; Pythagoras).
Use them to simplify — often by converting everything to sin and cos.
Prove identities by transforming one side into the other.
These identities make trig equations solvable in the next lesson.

How this fits the course

Build on

Trigonometric Ratios and Graphs

Addition and Double-Angle Formulae

Leads to

Solving Trigonometric Equations

Test yourself

Ready to lock in Trigonometric Identities? Pick a mode and earn XP & Dobloons.