If you’re taking a geometry or trigonometry class, one of the topics you’ll study are trigonometric identities. There are numerous trig identities, some of which are key for you to know, and others that you’ll use rarely or never. This guide explains the trig identities you should have memorized as well as others you should be aware of. We also explain what trig identities are and how you can verify trig identities.
In math, an "identity" is an equation that is always true, every single time. Trig identities are trigonometry equations that are always true, and they’re often used to solve trigonometry and geometry problems and understand various mathematical properties. Knowing key trig identities helps you remember and understand important mathematical principles and solve numerous math problems.
The 25 Most Important Trig Identities
Below are six categories of trig identities that you’ll be seeing often. Each of these is a key trig identity and should be memorized. It seems like a lot at first, but once you start studying them you’ll see that many follow patterns that make them easier to remember.
Basic Identities
These identities define the six trig functions.
$$sin(θ) = 1/{csc(θ)}$$
$$cos(θ) = 1/{sec(θ)}$$
$$tan(θ) = 1/{cot(θ)} = {sin(θ)}/{cos(θ)}$$
$$csc(θ) = 1/{sin(θ)}$$
$$sec(θ) = 1/{cos(θ)}$$
$$cot(θ) = 1/{tan(θ)} = {cos(θ)}/{sin(θ)}$$
Pythagorean Identities
These identities are the trigonometric proof of the Pythagorean theorem (that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, or $a^2 + b^2 = c^2$). The first equation below is the most important one to know, and you’ll see it often when using trig identities.
$$sin^2(θ) + cos^2(θ) = 1$$
$$tan^2(θ) + 1 = sec^2(θ)$$
$$1 + cot^2(θ) = csc^2(θ)$$
Co-function Identities
Each of the trig functions equals its co-function evaluated at the complementary angle.
$$sin(θ) = cos({π/2} - θ)$$
$$cos(θ) = sin({π/2} - θ)$$
$$tan(θ) = cot({π/2} - θ)$$
$$cot(θ) = tan({π/2} - θ)$$
$$csc(θ) = sec({π/2} - θ)$$
$$sec(θ) = csc({π/2} - θ)$$
Negative Angle Identities
Sine, tangent, cotangent, and cosecant are odd functions (symmetric about the origin). Cosine and secant are even functions (symmetric about the y-axis).
$$sin(-θ) = -sin(θ)$$
$$cos(-θ) = cos(θ)$$
$$tan(-θ) = -tan(θ)$$
Sum and Difference Identities
These are sometimes known as Ptolemy’s Identities as he’s the one who first proved them.
$$sin(α + β) = sin(α)cos(β) + cos(α)sin(β)$$
$$sin(α – β) = sin(α)cos(β) – cos(α)sin(β)$$
$$cos(α + β) = cos(α)cos(β) – sin(α)sin(β)$$
$$cos(α – β) = cos(α)cos(β) + sin(α)sin(β)$$
Double-Angle Identities
You only need to memorize one of the double-angle identities for cosine. The other two can be derived from the Pythagorean theorem by using the identity $sin^2(θ) + cos^2(θ) = 1$ to convert one cosine identity to the others.
$$sin(2θ) = 2 sin(θ) cos(θ)$$
$$cos(2θ) = cos^2(θ) – sin^2(θ) = 1 – 2 sin^2(θ) = 2 cos^2(θ) – 1$$
$$tan(2θ)={2 tan(θ)}/{1– tan^2(θ)}$$
Additional Trig Identities
These three categories of trig identities are used less often. You should look through them to make sure you understand them, but they typically don’t need to be memorized.
Half-Angle Identities
These are inversions of the double-angle identities.
$$sin2(θ) = {1/2}(1-cos (2θ))$$
$$cos2(θ) = {1/2}(1+ cos (2θ))$$
$$tan2(θ) = {1-cos(2θ)}/{1+ cos (2θ)}$$
Sum Identities
These trig identities make it possible for you to change a sum or difference of sines or cosines into a product of sines and cosines.
$$sin(α) + sin(β)= 2sin({α + β}/ 2) cos({α - β}/ 2)$$
$$sin(α) - sin(β)= 2cos({α + β}/ 2) sin({α - β}/ 2)$$
$$cos(α) + cos(β)= 2cos({α + β} / 2) cos({α - β}/ 2)$$
$$cos(α) - cos(β)= -2sin ({α + β}/ 2) sin({α - β}/ 2)$$
Product Identities
This group of trig identities allows you to change a product of sines or cosines into a product or difference of sines and cosines.
$$sin(α) cos(β)= {1/2}(sin (α + β) + sin (α - β))$$
$$cos(α) sin(β)= {1/2}(sin (α + β) - sin (α - β))$$
$$sin(α) sin(β)= {1/2}(cos (α - β) - cos(α + β))$$
$$cos(α) cos(β)= {1/2}(cos (α - β) + cos(α + β))$$
Verifying Trigonometric Identities
Once you have gone over all the key trig identities in your math class, the next step will be verifying them. Verifying trig identities means making two sides of a given equation identical to each other in order to prove that it is true. You’ll use trig identities to alter one or both sides of the equation until they’re the same.
Verifying trig identities can require lots of different math techniques, including FOIL, distribution, substitutions, and conjugations. Each equation will require different techniques, but there are a few tips to keep in mind when verifying trigonometric identities.
#1: Start With the Harder Side
Despite what you may initially want to do, we recommend starting with the side of the equation that looks messier or more difficult. Complicated-looking equations often give you more possibilities to try out than simpler equations, so start with the trickier side so you have more options.
#2: Remember That You Can Change Both Sides
You don’t need to stick to only changing one side of the equation. If you get stuck on one side, you can switch over to the other side and begin changing it as well. Neither side of the equation needs to be the same as how it was originally; as long as both sides of the equation end up being identical, the identity has been verified.
#3: Turn all the Functions Into Sines and Cosines
Most students learning trig identities feel most comfortable with sines and cosines because those are the trig functions they see the most. Make things easier on yourself by converting all the functions to sines and cosines!
Example 1
Verify the identity $cos(θ)sec(θ) = 1$
Let’s change that secant to a cosine. Using basic identities, we know $sec(θ) = 1/{cos(θ)}$. That gives us:
$$cos(θ) (1/{cos(θ)}) = 1$$
The cosines on the left cancel each other out, leaving us with $1=1$.
Identity verified!
Example 2
Verify the identity $1 − cos(2θ) = tan(θ) sin(2θ)$
Let’s start with the left side since it has more going on. Using basic trig identities, we know tan(θ) can be converted to sin(θ)/ cos(θ), which makes everything sines and cosines.
$$1 − cos(2θ) = ({sin(θ)}/{cos(θ)}) sin(2θ)$$
Distribute the right side of the equation:
$$1 − cos(2θ) = 2sin^2(θ)$$
There are no more obvious steps we can take to transform the right side of the equation, so let’s move to the left side. We can use the Pythagorean identity to convert $cos(2θ)$ to $1 - 2sin^2(θ)$
$$1 - (1 - 2sin^2(θ)) = 2sin^2(θ)$$
Now work out the left side of the equation
$$2sin^2(θ) = 2sin^2(θ)$$
The two sides are identical, so the identity has been verified!
Example 3
Verify the identity $sec(-θ) = sec(θ)$
The left side of the equation is a bit more complicated, so let’s change that secant into a sine or cosine. From the basic trig identities, we know that $sec(θ) = 1/{cos(θ)}$, which means that $sec(-θ) = 1/{cos(-θ)}$. Substitute that for the left side:
$$1/{cos(-θ)} = sec(θ)$$
The negative angle identities tell us that $cos(-θ) = cos(θ)$, so sub that:
$$1/{cos(θ)} = sec(θ)$$
Again, we know that $sec(θ) = 1/{cos(θ)}$, so we end up with:
$$sec(θ) = sec(θ)$$
Identity verified!
Summary: Trig Identities Solver
You’ll need to have key trig identities memorized in order to do well in your geometry or trigonometry classes. While there may seem to be a lot of trigonometric identities, many follow a similar pattern, and not all need to be memorized.
When verifying trig identities, keep the following three tips in mind:
- Start with the trickier side
- Remember that you can change both sides of the equation
- Turn the functions into sines and cosines
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