Visual walkthrough — Pythagorean identities — sin² + cos² = 1, derivations of the other two

We only assume you can read a number line for the $x$ (right/left) direction and one for the $y$ (up/down) direction. Everything else we build. See Unit Circle definition of sin and cos and Pythagoras Theorem for the two ideas we lean on.

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Step 1 — Draw the circle of radius 1

WHAT. We draw a circle centred at the point where the two number lines cross (the origin), and we make its radius exactly $1$ unit long.

WHY. A "radius $1$" circle is the simplest possible measuring tool: every point on its edge is the same distance — namely $1$ — from the centre. That fixed distance is the secret ingredient of the whole identity. We call it the unit circle.

PICTURE. Look at the red circle. Pick any point on its edge; a straight line from the centre to that point (the dashed red spoke) is always the same length, $1$.

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Step 2 — Name an angle, and name the point it lands on

WHAT. Start at the rightmost point of the circle (straight along the positive $x$-axis). Now sweep anticlockwise by some amount. Call the amount you swept $\theta$ (the Greek letter "theta", just a name for "the angle"). The spoke now points at one particular edge-point $P$.

WHY. We need a way to say which point on the circle we mean. The angle $\theta$ does that: one angle picks out exactly one point.

PICTURE. The red wedge is the angle $\theta$, opening from the $x$-axis up to the spoke. The red dot at the spoke's tip is our point $P$.

$P=(\underset{\text{how far right}}{\underbrace{x}},\underset{\text{how far up}}{\underbrace{y}})$

Right now $x$ and $y$ are just "the coordinates of $P$." In the next step we give them their famous names.

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Step 3 — Define $\cos \theta$ and $\sin \theta$ as those coordinates

WHAT. We define two new words: $\cos \theta =x\text{-coordinate of }P,\sin \theta =y\text{-coordinate of }P.$

WHY. This is a definition, not a result — nothing to prove. We are just agreeing that "cosine of $\theta$" is shorthand for "the sideways position of the point at angle $\theta$", and "sine of $\theta$" for "the upward position." (This is the exact meaning used in Unit Circle definition of sin and cos.)

PICTURE. The red horizontal segment along the bottom is $\cos \theta$. The red vertical segment going up to $P$ is $\sin \theta$. They are literally the two coordinates of the dot.

$P=(\stackrel{\text{sideways}}{\overbrace{\cos \theta }},\stackrel{\text{upward}}{\overbrace{\sin \theta }})$

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Step 4 — Build the right triangle

WHAT. Drop a straight vertical line from $P$ down to the $x$-axis. This creates a triangle with three sides: a flat bottom, an upright side, and the slanted spoke.

WHY. We are hunting for a relationship between $\cos \theta$ and $\sin \theta$. Triangles with a square corner (a right angle, a perfect $90^{\circ }$ corner) obey Pythagoras' theorem — a rock-solid rule connecting their three sides. So if we can make a right triangle out of our picture, Pythagoras will hand us a relationship for free.

PICTURE. The little red square marks the $90^{\circ }$ corner where the vertical drop meets the $x$-axis. That corner is what makes this a right triangle.

• Bottom side length $=|\cos \theta |$ (its horizontal run).
• Upright side length $=|\sin \theta |$ (its vertical rise).
• Slanted side (the spoke) length $=1$ (the radius).

The bars $|\cdot |$ mean absolute value — "distance, always a positive number." A length can't be negative even when $\cos \theta$ itself is negative (as it is on the left side of the circle). We handle that carefully in Step 6.

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Step 5 — Apply Pythagoras and get the master identity

WHAT. Pythagoras' theorem says: for a right triangle, (one leg)² + (other leg)² = (slanted side)². We plug in our three sides.

WHY. This is the single algebra step that forces the identity. We use Pythagoras and not some other rule because it is precisely the tool that links the two legs to the hypotenuse — exactly the two quantities ($\cos \theta$, $\sin \theta$) we want to tie together.

PICTURE. The two red legs squared, added, equal the black hypotenuse squared. The hypotenuse is $1$, and $1^2=1$, so the right-hand side collapses to just $1$.

$\underset{{\text{bottom leg}}^2}{\underbrace{|\cos \theta {|}^2}}+\underset{{\text{upright leg}}^2}{\underbrace{|\sin \theta {|}^2}}=\underset{{\text{hypotenuse}}^2}{\underbrace{1^2}}$

Squaring throws away the absolute-value bars, because any number squared is positive: $|\cos \theta {|}^2={\cos }^2\theta$ and $|\sin \theta {|}^2={\sin }^2\theta$. That is the quiet magic — squaring makes the sign irrelevant. We land on:

Here ${\cos }^2\theta$ is short for $(\cos \theta {)}^2$ — square the number $\cos \theta$. It does not mean $\cos ({\theta }^2)$.

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Step 6 — Check EVERY quadrant (the sign-safety step)

WHAT. We just derived the identity using a triangle in the top-right region. Does it still hold when $P$ is on the left, or below, where $\cos \theta$ or $\sin \theta$ turn negative?

WHY. A derivation is only trustworthy if it survives every case. The four regions of the plane are called quadrants (Q1 top-right, Q2 top-left, Q3 bottom-left, Q4 bottom-right). We must confirm no quadrant breaks the identity.

PICTURE. Four red dots, one per quadrant. In each, the signs of $(\cos \theta ,\sin \theta )$ are labelled. Notice the triangle's legs are the same length in every quadrant — only the direction (sign) flips.

Quadrant	$\cos \theta$	$\sin \theta$	${\cos }^2\theta +{\sin }^2\theta$
Q1 (top-right)	$+$	$+$	$(+{)}^2+(+{)}^2=1$
Q2 (top-left)	$-$	$+$	$(-{)}^2+(+{)}^2=1$
Q3 (bottom-left)	$-$	$-$	$(-{)}^2+(-{)}^2=1$
Q4 (bottom-right)	$+$	$-$	$(+{)}^2+(-{)}^2=1$

Because squaring destroys the minus sign, the sum is $1$ everywhere. The identity is quadrant-proof.

Degenerate cases — where the triangle collapses to a line — also pass:

$\theta$	$P$	$\cos \theta$	$\sin \theta$	sum
$0^{\circ }$	$(1,0)$	$1$	$0$	$1^2+0^2=1$
$90^{\circ }$	$(0,1)$	$0$	$1$	$0^2+1^2=1$
$180^{\circ }$	$(-1,0)$	$-1$	$0$	$1$
$270^{\circ }$	$(0,-1)$	$0$	$-1$	$1$

At these four points one leg has length $0$ — the triangle flattens — yet Pythagoras still reads correctly.

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Step 7 — Divide the picture to grow the two children

WHAT. Take the master identity and divide every term by ${\cos }^2\theta$. Then do it again dividing by ${\sin }^2\theta$.

WHY. We want identities involving $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$ (and their $\sin$-based cousins). Those fractions appear the instant we divide by ${\cos }^2\theta$ — so division is the natural, targeted move. These new words $\tan ,\sec ,\cot ,\csc$ come from Reciprocal and Quotient trig identities.

PICTURE. The master identity sits at the top; two red arrows fan down to the two children. Each arrow is labelled with what we divided by.

Dividing by ${\cos }^2\theta$ (allowed only when $\cos \theta \ne 0$): $\underset{{\tan }^2\theta }{\underbrace{\frac{{\sin }^2\theta }{{\cos }^2\theta }}}+\underset{1}{\underbrace{\frac{{\cos }^2\theta }{{\cos }^2\theta }}}=\underset{{\sec }^2\theta }{\underbrace{\frac{1}{{\cos }^2\theta }}}⟹\boxed{1+{\tan }^2\theta ={\sec }^2\theta }$

Dividing by ${\sin }^2\theta$ (allowed only when $\sin \theta \ne 0$): $\underset{1}{\underbrace{\frac{{\sin }^2\theta }{{\sin }^2\theta }}}+\underset{{\cot }^2\theta }{\underbrace{\frac{{\cos }^2\theta }{{\sin }^2\theta }}}=\underset{{\csc }^2\theta }{\underbrace{\frac{1}{{\sin }^2\theta }}}⟹\boxed{1+{\cot }^2\theta ={\csc }^2\theta }$

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The one-picture summary

One point $P$ on the unit circle → its coordinates are $(\cos \theta ,\sin \theta )$ → drop a perpendicular to make a right triangle → Pythagoras gives ${\sin }^2\theta +{\cos }^2\theta =1$ → divide by ${\cos }^2\theta$ or ${\sin }^2\theta$ to spawn the two children. That single red spoke of length $1$ is the source of all three identities.

Recall Feynman retelling — say it to a friend

"Draw a circle one metre across from the middle. Walk out to the edge in any direction. How far right you moved, call cos; how far up, call sin. Since you're always one metre from the middle, the corner-triangle rule (right²+up²=slant²) says cos² plus sin² equals one — always, in every direction, even when you walk left or down, because squaring hides the minus signs. Then if you shrink the whole picture by dividing by cos², you get '1 plus tan² is sec²'; divide by sin² instead and you get '1 plus cot² is csc².' Same one picture, three sentences."

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Connections

• Parent: Pythagorean identities (parent)
• Unit Circle definition of sin and cos
• Pythagoras Theorem
• Reciprocal and Quotient trig identities
• Trig identity proofs — strategy
• Double angle formulas
• Solving trig equations