# Trig identity reference

Look up AND understand all your favorite trig identities.

## Reciprocal and quotient identities

$\sec(\theta)= \dfrac{1}{\cos(\theta)}$
Play with point on the unit circle to see how cosine and secant change together. Notice how when cosine is small, secant is big, and vice versa. It turns out they always multiply to exactly 1.
We can also see this identity using similar triangles. Slide the dot below the graph to see one triangle transform into the other. Watch carefully to see which segments correspond to each other.

$\csc(\theta)= \dfrac{1}{\sin(\theta)}$
Play with point on the unit circle to see how sine and cosecant change together. Notice how when sine is small, cosecant is big, and vice versa. It turns out they always multiply to exactly 1.
We can also see this identity using similar triangles. Slide the dot below the graph to see one triangle transform into the other. Watch carefully to see which segments correspond to each other.

cotangent, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, tangent, left parenthesis, theta, right parenthesis, end fraction
Play with point on the unit circle to see how tangent and cotangent change together. Notice how when tangent is small, cotangent is big, and vice versa. It turns out they always multiply to exactly 1.
We can also see this identity using similar triangles. Slide the dot below the graph to see one triangle transform into the other. Watch carefully to see which segments correspond to each other.

tangent, left parenthesis, theta, right parenthesis, equals, start fraction, sine, left parenthesis, theta, right parenthesis, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction
We can see this identity using similar triangles. Slide the dot below the graph to see one triangle transform into the other. Watch carefully to see which segments correspond to each other.

cotangent, left parenthesis, theta, right parenthesis, equals, start fraction, cosine, left parenthesis, theta, right parenthesis, divided by, sine, left parenthesis, theta, right parenthesis, end fraction
We can see this identity using similar triangles. Slide the dot below the graph to see one triangle transform into the other. Watch carefully to see which segments correspond to each other.

## Pythagorean identities

sine, start superscript, 2, end superscript, left parenthesis, theta, right parenthesis, plus, cosine, start superscript, 2, end superscript, left parenthesis, theta, right parenthesis, equals, 1, start superscript, 2, end superscript
This identity comes from writing down the Pythagorean theorem for the triangle below.
$\tan^2(\theta) + 1^2=\sec^2(\theta)$
This identity comes from writing down the Pythagorean theorem for the triangle below.
$\cot^2(\theta) + 1^2=\csc^2(\theta)$
This identity comes from writing down the Pythagorean theorem for the triangle below.

## Identities that come from sums, differences, multiples, and fractions of angles

These are all closely related, but let's go over each kind.
Angle sum and difference identities
$\sin(\theta+\phi)=\sin\theta\cos\phi+\cos\theta\sin\phi$
$\sin(\theta-\phi)=\sin\theta\cos\phi-\cos\theta\sin\phi$
$\cos(\theta+\phi)=\cos\theta\cos\phi-\sin\theta\sin\phi$
$\cos(\theta-\phi)=\cos\theta\cos\phi+\sin\theta\sin\phi$
The figure below shows a way of proving the angle sum identities. The left and right sides of the rectangle are equal, giving us:
$\purple{\sin(\theta+\phi)}=\blue{\sin\theta}\red{\cos\phi}+\blue{\cos\theta}\red{\sin\phi}$
The top and bottom sides are also equal, giving us:
$\purple{\cos(\theta+\phi)}=\blue{\cos\theta}\red{\cos\phi}-\blue{\sin\theta}\red{\sin\phi}$
It's easiest to make sense of this diagram if you start with the right triangle in the middle of the diagram and build outward using the right triangle definitions of the trig functions (SOH CAH TOA).
A similar diagram could show the angle difference identities. Can you draw one?
(Technically, this isn't a proof for all possible values of theta and $\phi$, but the identities do apply for all values.)
$\tan(\theta+\phi)=\dfrac{\tan\theta+\tan\phi}{1-\tan\theta\tan\phi}$
$\tan(\theta-\phi)=\dfrac{\tan\theta-\tan\phi}{1+\tan\theta\tan\phi}$
The figure below shows a way of proving the angle sum identity for tangent.
This one is a little bit tricky to make sense of. It's all about building up to the triangle in the upper-left part of the diagram. When you apply the right triangle definition of tangent to that triangle, you get:
It's easiest to make sense of this diagram if you start with the segment at the bottom of the diagram and build upward using the right triangle definitions of the trig functions (SOH CAH TOA).
A similar diagram could show the angle difference identity. Can you draw one?
(Technically, this isn't a proof for all possible values of theta and $\phi$, but the identities do apply for all values.)
Double angle identities
sine, left parenthesis, 2, theta, right parenthesis, equals, 2, sine, theta, cosine, theta
We can get this identity if we take the angle sum identity for sine, but make both angles the same.
$\sin(\blue\theta+\red\phi)=\sin\blue\theta\cos\red\phi+\cos\blue\theta\sin\red\phi$
sine, left parenthesis, start color blue, theta, end color blue, plus, start color blue, theta, end color blue, right parenthesis, equals, sine, start color blue, theta, end color blue, cosine, start color blue, theta, end color blue, plus, cosine, start color blue, theta, end color blue, sine, start color blue, theta, end color blue
sine, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, 2, sine, start color blue, theta, end color blue, cosine, start color blue, theta, end color blue
cosine, left parenthesis, 2, theta, right parenthesis, equals, 2, cosine, start superscript, 2, end superscript, theta, minus, 1
We can also get this one from the angle sum identity, but we need to do a little extra manipulation.
First, let's make both angles the same.
$\cos(\blue\theta+\red\phi)=\cos\blue\theta\cos\red\phi-\sin\blue\theta\sin\red\phi$
cosine, left parenthesis, start color blue, theta, end color blue, plus, start color blue, theta, end color blue, right parenthesis, equals, cosine, start color blue, theta, end color blue, cosine, start color blue, theta, end color blue, minus, sine, start color blue, theta, end color blue, sine, start color blue, theta, end color blue
cosine, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, cosine, start superscript, 2, end superscript, start color blue, theta, end color blue, minus, sine, start superscript, 2, end superscript, start color blue, theta, end color blue
Now, we can use the identity sine, start superscript, 2, end superscript, theta, plus, cosine, start superscript, 2, end superscript, theta, equals, 1 to put the right-hand side in terms of just cosines.
cosine, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, cosine, start superscript, 2, end superscript, start color blue, theta, end color blue, minus, left parenthesis, 1, minus, cosine, start superscript, 2, end superscript, start color blue, theta, end color blue, right parenthesis
cosine, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, cosine, start superscript, 2, end superscript, start color blue, theta, end color blue, minus, 1, plus, cosine, start superscript, 2, end superscript, start color blue, theta, end color blue
cosine, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, 2, cosine, start superscript, 2, end superscript, start color blue, theta, end color blue, minus, 1
tangent, left parenthesis, 2, theta, right parenthesis, equals, start fraction, 2, tangent, theta, divided by, 1, minus, tangent, start superscript, 2, end superscript, theta, end fraction
This one also comes from the angle sum identity.
$\tan(\blue\theta+\red\phi)=\dfrac{\tan\blue\theta+\tan\red\phi}{1-\tan\blue\theta\tan\red\phi}$
tangent, left parenthesis, start color blue, theta, end color blue, plus, start color blue, theta, end color blue, right parenthesis, equals, start fraction, tangent, start color blue, theta, end color blue, plus, tangent, start color blue, theta, end color blue, divided by, 1, minus, tangent, start color blue, theta, end color blue, tangent, start color blue, theta, end color blue, end fraction
tangent, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, start fraction, 2, tangent, start color blue, theta, end color blue, divided by, 1, minus, tangent, start superscript, 2, end superscript, start color blue, theta, end color blue, end fraction
Half angle identities
sine, start fraction, theta, divided by, 2, end fraction, equals, plus minus, square root of, start fraction, 1, minus, cosine, theta, divided by, 2, end fraction, end square root
cosine, start fraction, theta, divided by, 2, end fraction, equals, plus minus, square root of, start fraction, 1, plus, cosine, theta, divided by, 2, end fraction, end square root
\begin{aligned} \tan\dfrac{\theta}{2}&=\pm\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}}\\ \\ &=~~~~~~~\dfrac{1-\cos\theta}{\sin\theta}\\ \\ &=~~~~~~~\dfrac{\sin\theta}{1+\cos\theta}\end{aligned}
We can get all of these by taking the double angle formulas and making the substitution:
start color blue, theta, end color blue, right arrow, start color red, start fraction, theta, divided by, 2, end fraction, end color red
Then there's some rearranging to do.
For example:
sine, left parenthesis, 2, start color blue, theta, end color blue, right parenthesis, equals, 2, sine, start color blue, theta, end color blue, cosine, start color blue, theta, end color blue
turns into...
And then we need to solve for sine, start color red, start fraction, theta, divided by, 2, end fraction, end color red. Give it a shot, and see if you can figure out how to show the other identities.

## Symmetry and periodicity identities

sine, left parenthesis, minus, theta, right parenthesis, equals, minus, sine, left parenthesis, theta, right parenthesis
We can see this by looking at a unit circle diagram.
cosine, left parenthesis, minus, theta, right parenthesis, equals, plus, cosine, left parenthesis, theta, right parenthesis
We can see this by looking at a unit circle diagram.
tangent, left parenthesis, minus, theta, right parenthesis, equals, minus, tangent, left parenthesis, theta, right parenthesis
This one is harder to see on a unit circle diagram, but we can get it by writing tangent in terms of sine and cosine, then applying the sine and cosine identities for negative angles.
tangent, left parenthesis, theta, right parenthesis, equals, start fraction, sine, left parenthesis, theta, right parenthesis, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction
So,
tangent, left parenthesis, minus, theta, right parenthesis, equals, start fraction, sine, left parenthesis, minus, theta, right parenthesis, divided by, cosine, left parenthesis, minus, theta, right parenthesis, end fraction, equals, start fraction, minus, sine, left parenthesis, theta, right parenthesis, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction, equals, minus, tangent, left parenthesis, theta, right parenthesis

sine, left parenthesis, theta, plus, 2, pi, right parenthesis, equals, sine, left parenthesis, theta, right parenthesis
cosine, left parenthesis, theta, plus, 2, pi, right parenthesis, equals, cosine, left parenthesis, theta, right parenthesis
tangent, left parenthesis, theta, plus, pi, right parenthesis, equals, tangent, left parenthesis, theta, right parenthesis

## Cofunction identities

sine, theta, equals, cosine, left parenthesis, start fraction, pi, divided by, 2, end fraction, minus, theta, right parenthesis
cosine, theta, equals, sine, left parenthesis, start fraction, pi, divided by, 2, end fraction, minus, theta, right parenthesis
tangent, theta, equals, cotangent, left parenthesis, start fraction, pi, divided by, 2, end fraction, minus, theta, right parenthesis
c, o, t, theta, equals, tangent, left parenthesis, start fraction, pi, divided by, 2, end fraction, minus, theta, right parenthesis
$\sec\theta= \csc(\dfrac{\pi}{2}-\theta)$
$\csc\theta= \sec(\dfrac{\pi}{2}-\theta)$
All these identities look the same!
The key to understanding these identities is realizing that going from theta to start fraction, pi, divided by, 2, end fraction, minus, theta is equivalent to reflecting your angle over the line y, equals, x.
The interactive diagram below attempts to show how the angles relate to each other and that start color red, sine, theta, end color red, equals, start color blue, cosine, left parenthesis, start fraction, pi, divided by, 2, end fraction, minus, theta, right parenthesis, end color blue. The other co-function pairs work the same way, leading to the other identities.

## Appendix: All trig ratios in the unit circle

Use the movable point to see how the lengths of the ratios change according to the angle.