Trigonometry is abundant with identities—they provide important renaming tools
when working with trigonometric expressions.
Three of the most basic trigonometric identities are
discussed in this section, after a quick review of the word ‘identity’.
What is an ‘identity’?
An identity is a mathematical sentence that is always true.
Strictly speaking, ‘1 + 1 = 2’ is an identity.
However, the word ‘identity’ is typically reserved for a sentence with one or more variables, that is true
for every possible choice of variable(s).
For example, $\,(x+y)^2 = x^2 + 2xy + y^2\,$ is an identity from algebra.
No matter what real numbers are chosen for $\,x\,$ and $\,y\,$, the equation is true.
For example, letting $\,x = 2\,$ and $\,y = 3\,$ gives:
 left side: $(x+y)^2 = (23)^2 = (1)^2 = 1$
 right side: $x^2 + 2xy + y^2 = 2^2 + 2(2)(3) + (3)^2 = 4  12 + 9 = 1$
The left and right sides will
always be equal,
regardless of the choices made
for $\,x\,$ and $\,y\,$.
In this case, it is a simple application of
FOIL to see why the equation is always true:
$$(x+y)^2 = (x+y)(x+y) = x^2 + 2xy + y^2$$
However, identities are not always this obvious—it is not always this easy to prove that a given sentence is an identity!
What good are identities?
Identities can provide tremendous renaming power.
For example, the identity above allows the expressions $\,(x+y)^2\,$ and $\,x^2 + 2xy + y^2\,$
to be substituted,
one for the other, whenever it is convenient to do so.
In particular, renaming $\,x^2 + 2xy + y^2\,$ as $\,(x+y)^2\,$ (a perfect square) shows that
it is always nonnegative.
The sum $\,x^2 + 2xy + y^2\,$ doesn't readily reveal that it can't ever be negative—but the perfect square $\,(x+y)^2\,$ does.
Different names can reveal different properties of numbers!
The Pythagorean Identity: $\,\sin^2 t + \cos^2 t = 1\,$
The Pythagorean Identity, $\,\sin^2 t + \cos^2 t = 1\,$, is perhaps the most used and most famous trigonometric identity.
Remember—when a mathematical result is given a special name, there's a reason!
The Pythagorean Identity follows immediately from the unit circle definition of sine and cosine:

Recall that, in trigonometry, ‘unit circle’ refers to the circle of radius $\,1\,$ that is centered at the origin.
The equation of the unit circle is: $x^2 + y^2 = 1$
 By definition, cosine and sine give the $x$ and $y$values (respectively) of points on the unit circle.
That is, for every real number $\,t\,$ (which can be thought of as the radian measure of an angle, if desired),
$\,\bigl(\cos t,\sin t\bigr)\,$ is a point on the unit circle.
 Since $\,\bigl(\cos t,\sin t\bigr)\,$ is on the circle $\,x^2 + y^2 = 1\,$, it satisfies the equation.
That is, substitution of ‘$\,\cos t\,$’ for ‘$\,x\,$’ and ‘$\,\sin t\,$’ for ‘$\,y\,$’ makes the equation true:
$$
(\cos t)^2 + (\sin t)^2 = 1
$$
 The expression ‘$\,\sin^2 t\,$’ is a common abbreviation for ‘$\,(\sin t)^2\,$’.
(You save writing two parentheses!)
Similarly, ‘$\,\cos^2 t\,$’ is a common abbreviation for ‘$\,(\cos t)^2\,$’.
(More generally, this abbreviation is also used for powers $\,3, 4, 5, \ldots\,$.)
 Using these abbreviations and switching the order of the sum gives the Pythagorean Identity:
$$
\sin^2 t + \cos^2 t = 1
$$


Why the name ‘the Pythagorean Identity’ ?
Look at the green triangle shown at right, in the first quadrant, in the unit circle:
 the bottom leg has length $\,\cos t\,$
 the other leg has length $\,\sin t\,$
 the hypotenuse has length $\,1\,$
A quick application of the Pythagorean Theorem gives:
$$
\sin^2 t + \cos^2 t = 1^2 = 1
$$
Voila!
The Pythagorean Identity!


Cosine is an Even Function
Recall that even functions have the property that when inputs are opposites,
outputs are the same:
 The numbers $\,t\,$ and $\,t\,$ are opposites, for all real numbers $\,t\,$.
 Their corresponding outputs from a function $\,f\,$ are $\,f(t)\,$ and $\,f(t)\,$.
 For even functions, these two outputs must (always) be equal: $\,f(t) = f(t)\,$.
Cosine has this property:


By definition:
 the terminal point for $\,t\,$ has $x$value equal to $\,\cos(t)\,$
 the terminal point for $\,\color{red}{t}\,$
has $\color{red}{x}$value equal to $\,\color{red}{\cos(t)}\,$
From symmetry, these two $x$values are always the same!

Sine is an Odd Function
Recall that odd functions have the property that when inputs are opposites,
outputs are also opposites:
 The numbers $\,t\,$ and $\,t\,$ are opposites, for all real numbers $\,t\,$.
 Their corresponding outputs from a function $\,f\,$ are $\,f(t)\,$ and $\,f(t)\,$.
 For odd functions, these two outputs must (always) be opposites: $\,f(t) = f(t)\,$.
Sine has this property:


By definition:
 the $\,t\,$ has $y$value equal to $\,\sin(t)\,$
 the terminal point for
$\,\color{red}{t}\,$
has
$\color{red}{y}$value equal to
$\,\color{red}{\sin(t)}\,$
From symmetry, these two $y$values are always opposites:
$$
\overbrace{\strut\sin(t)}^{\text{sine of $t$}}\ \ \
\overbrace{\strut =}^{\text{is}}\ \ \
\overbrace{\strut }^{\text{the opposite of}}\ \ \
\overbrace{\strut\sin(t)}^{\text{sine of $t$}}
$$

Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Periodic Functions