Trigonometric Identities Explained
Trigonometric identities can be very useful in simplifying seemingly complex trigonometric expressions and equations into much more manageable forms. They may seem daunting and imposing at first, but they really boil down to just a few simple formulas. In this guide, you’ll learn these identities along with clear and easy-to-follow examples of their practical use.
What Is a Trigonometric Identity?
Trigonometric identities are equalities that involve trigonometric functions that are true for every single value of the occurring variables. To put that in other words, they are equations that hold true regardless of the values of the angles being chosen.
Trigonometric identities serve a purpose of not only defining the groundwork for trigonometry as a subject, but also for helping one simplify and solve what would otherwise be challenging problems. By utilizing identities, a complex trigonometric equation can be converted into a form that’s easier to digest via standard algebra tools.
Reciprocal Trigonometric Identities
Reciprocal trigonometric identities involve the relation between the core six trigonometric functions: sine, cosine, tangent, secant, cosecant, and the cotangent. These identities express each function via a formula that makes it equal to one over another trig function.
The six identities are:
Thus, the reciprocal identities can simplify equations that involve multiple trigonometric functions into ones consisting of only one function type. For example, the expression 
can be converted into a much simpler formula. Since 
, we can rewrite the expression as 
which can be further simplified to 
. This can in turn be further reduced to 
via the rules governing the addition of fractions. We could have also alternatively used the 
identity and obtained a result in terms of the cosecant, which would be 
.
Pythagorean Trigonometric Identities
Pythagorean trigonometric identities, as their name suggests, were discovered by the Greek mathematician Pythagoras and also represent equations that hold true regardless of the angle in question. The three core identities are:
These identities can also be utilized to help simplify trigonometric expressions and equations. For example, let’s utilize the first Pythagorean identity so simplify the expression 
. If we break apart the first term, we can see that the rest of the expression holds the trigonometric identity: 
. Working on it further, we obtain 
which at the end becomes just 
since the sum of the sine squared and cosine squared equals one.
Utilizing the second and third identities yields similar benefits. For example, we can utilize the second Pythagorean identity in combination with the reciprocal identity from the first section of this guide to simplify 
into 
and further into 
which is equal to 1.
Ratio Trigonometric Identities
Ratio identities represent relations in the ratios of core trigonometric functions, namely the sine and the cosine. They are alternatively labeled quotient trigonometric identities since the quotient of the sine and the cosine functions is utilized to derive the functions for the tangent and the cotangent. Quite simply, the identities are:
Let’s use these identities to once again simplify complex trigonometric expression into much more manageable ones. For example, consider the daunting-looking 
. We can use the ratio identities to break down the cotangent and the tangent terms into their sine and cosine components. In applying the identities to the above expression, we obtain 
. The expression looks even “bigger” now, but the path to simplicity is much more clear—we simply apply the rules of division and reduce the terms in the fraction. On the left side, the cosine squares cancel each other out and on the right side, the sin squared terms also reduce down to one. Thus, we obtain a much simpler expression of 
which, as we know already, equals to just one. Quite a big difference between that and the original scary-looking expression, isn’t it?
Double Angle Trigonometric Identities
Double angle identities take a departure from the aforementioned trigonometric identities in the sense that they help deal with functions where the argument is not just Θ but 2Θ or other even multiples of it. Thus, an expression such involving elements such as 
can be broken down into simpler components we are used to working with before: functions of the sine, cosine, and tangent.
Here are the three main double angle identities:
, which can be alternatively expressed as 
or 
As with the previous identities, let’s see it an action and apply the formula to help us simplify trigonometric expressions. Suppose we want to reduce 
. We can apply the trigonometric identity for alternatively expressing 
and obtain 
which conveniently further reduces to 
.
Make sure to not confuse 
and 
, they are very different expressions! For example, if we take Θ to be equal 45°, then 
equals 
, which is equal to 1 versus 
which equals 
or, alternatively, ~1.414.
Let’s try another example with the double angle formula for cosine. Suppose we try to simplify 
. If we take one of the forms of the double angle formula for the cosine, the term 
can be expressed as 
resulting in our original expression being equal to 
. In simplifying the numerator, we obtain 
which further reduces to 
once we apply the ratio identity for the cotangent.
With a little practice, trigonometric identities can become trusted members of your mathematical “toolbox.” And remember, if one approach to simplification does not give you the result you’re looking for, keep trying with other trigonometric identities; eventually a skilled application of the needed formula will produce the simple expression needed for the solution. Happy simplifying!
by Izolda Fotiyeva, Ph.D., and Dmitriy Fotiyeva, authors of The Complete Idiot’s Guide to Trigonometry
