Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofs: Foundations

This page assumes you have seen none of the notation in the parent note. We build every symbol from the ground up. Read top to bottom; nothing below uses a symbol defined further down.

1. Angle — a measured amount of turning

Two ways to write "how much":

Degrees: a full turn is $360°$ . A quarter turn is $90°$ .
Radians: a full turn is $2 π$ . A quarter turn is $\frac{π}{2}$ .

Figure — Sum and difference formulas — sin(A±B), cos(A±B), tan(A±B) — proofs

The letters $A$ and $B$ in the parent note are just names for two such angles — two separate amounts of turning we will later combine.

2. The unit circle — where angles become points

Why radius one? Because then the point's coordinates are the sine and cosine directly, with no scaling to undo. This is the single most important simplification in all of trigonometry.

Pick an angle $θ$ (Greek letter "theta", a generic name for an angle). Turn a ray by $θ$ from the right-pointing direction. Where the ray crosses the circle is one specific point. Every angle names exactly one such point — that is the bridge from turning to geometry the parent topic is built on.

3. Coordinates $(x, y)$ — right-ness and up-ness

The origin is $(0, 0)$ . The point straight right on the unit circle is $(1, 0)$ ; straight up is $(0, 1)$ ; straight left is $(- 1, 0)$ ; straight down is $(0, - 1)$ .

4. Cosine and sine — the two numbers of a point

All four quadrants (the topic needs every one):

Region	angle range	$cos θ$	$sin θ$
Quadrant I (up-right)	$0$ to $90°$	$+$	$+$
Quadrant II (up-left)	$90°$ to $180°$	$-$	$+$
Quadrant III (down-left)	$180°$ to $270°$	$-$	$-$
Quadrant IV (down-right)	$270°$ to $360°$	$+$	$-$

At the boundaries (degenerate cases): $cos 0 = 1, sin 0 = 0$ (point $(1, 0)$ ); $cos 90° = 0, sin 90° = 1$ (point $(0, 1)$ ); $cos 180° = - 1, sin 180° = 0$ ; $cos 270° = 0, sin 270° = - 1$ . Notice each is $\pm 1$ or $0$ — the axis crossings.

See the full companion topic Unit Circle and Trig Definitions.

5. Tangent — the slope of the ray

6. Squares and the Pythagorean identity

This is the "collapse trick" the parent uses in Step 2 to turn four squared terms into the clean number $2$ . Full detail: Pythagorean Identity.

7. Even and odd — what a negative angle does

Why the topic needs this: the parent gets $cos (A + B)$ from $cos (A - B)$ for free by swapping $B \to - B$ . That swap only works because we know exactly how the minus travels through cosine (does nothing) and sine (flips). See also Cofunction Identities.

8. The distance formula — measuring a chord

The parent's whole proof rests on measuring the chord (straight line between two circle points) two different ways using this. Squaring avoids square roots and keeps the algebra clean.

9. Reading the parent's symbols

Recall What does

$P = (cos A, sin A)$ mean now? The point you land on after turning by angle $A$ from the right-pointing direction, on the unit circle. Its right-ness is $cos A$ , its up-ness is $sin A$ .

Recall What does the

$\pm$ / $\mp$ pair mean in $tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}$ ? Read them together: choose the top sign of each pair for the $+$ case, the bottom sign for the $-$ case. When numerator uses $+$ , denominator uses $-$ , and vice versa.

Prerequisite map

Equipment checklist

Self-test: cover the right side and answer each before revealing.

An angle measures what physical thing?

An amount of turning (rotation) from the positive

x

-axis, anticlockwise.

A full turn is how many degrees? Radians?

360°

and

2 π

Why is the circle chosen with radius

1

So a point's coordinates are

cos θ

and

sin θ

with no scaling.

cos θ

is which coordinate of the point?

The

x

-coordinate (right-ness).

sin θ

is which coordinate?

The

y

-coordinate (up-ness).

In Quadrant III, what are the signs of

cos

and

sin

Both negative.

Define

tan θ

as a ratio.

sin θ / cos θ

— up-ness over right-ness, the ray's slope.

When is

tan θ

undefined and why?

90°

and

270°

, where

cos θ = 0

(division by zero, vertical ray).

State the Pythagorean identity.

cos^{2} θ + sin^{2} θ = 1

What does

cos^{2} θ

mean?

(cos θ)^{2}

— cosine first, then squared.

What does

- θ

do to a point geometrically?

Reflects it across the

x

-axis (turn the other way).

cos (- θ) = ?

and

sin (- θ) = ?

cos θ

(even) and

- sin θ

(odd).

Write the squared distance between

(x_{1}, y_{1})

and

(x_{2}, y_{2})

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Connections

Parent topic — everything here feeds the six identities.
Unit Circle and Trig Definitions — the source of $(cos θ, sin θ)$ .
Pythagorean Identity — the Step-2 collapse trick.
Cofunction Identities — even/odd and the $\frac{π}{2} - θ$ bridge.
Double Angle Formulas — what you unlock next.
Product-to-Sum and Sum-to-Product — built by adding these formulas.
Derivatives of Trig Functions — where radians become essential.