Tuesday, March 06, 2012

Trigonometry: Proving Trigonometric Identities

**Notice:** This post was updated on March 10, 2015 to reflect a correction in the Trigonometric Identities image. The accompanying Google Document was also corrected, as was the link to that Document. If you downloaded the Google Document before 03/10/2015, please re-visit this link and download the new version. Thank you, and sorry for the inconvenience.

CLICK HERE to view the YouTube video that corresponds to this post.
Identity problems are probably the toughest part of Trigonometry for most people. They can look scary and huge and daunting, and many students become overwhelmed at just the sight of them. I hope that this post will make you less fearful of identity problems.

The first thing to address is that you need to know the trig identities. Just knowing these will make the problems half as challenging. The best way to learn them is to use them. Have a sheet with all of the identities listed (so that you don't have to flip through your book) and use this sheet only when you need it. If you think you know the identity to use, go ahead and use it and see what you get. Use your "cheat sheet" as a safety net, not as a life preserver.


I've created a small list of the main identities (reciprocal, Pythagorean, sum/difference, double-angle, and half-angle) in a document that you can print for free. You can then fold the page in half (or cut it) and keep it in your notebook as a handy reference card. You can view/print the page by following this link to the Google Document (opens in new window). Here it is below as an image for reference in this post.



The second thing you need to do is find out which identities you are required to have memorized for your test/exam. Ask your instructor which, if any, identities will be given to you, and which you are required to know ahead of time. Take the time to study the identities that will not be given to you. My two favorite ways to study this type of thing is to (1) make flashcards or (2) write them out without looking at the cheat sheet over and over again, checking myself each time. Writing is a wonderful tool for memorization!Now, assuming you have your cheat sheet handy, let's dig in to identity problems.

There are not many strategies for solving identity problems/proofs. The goal is always to show that the two sides are equal. There are three ways to accomplish this.1. Begin with the left side and use known trig identities to rewrite it until it looks like the right side.

2. Begin with the right side and use known trig identities to rewrite it until it looks like the left side.

3. Rewrite either side at a time until they are equal to each other.

I would stick to the first two methods. Always choose one side and stick with it for a while. If you decide it's not working out or you get stuck, go back to the beginning and start with the other side. If you work on both sides at a time, you are more likely to become confused and frustrated. So, how do you choose a side? Here are the things I look for:

1. Complexity -- It is easier to simplify a complicated side than to complicate a simple side. So look for the side that is "bigger" or more complex.


2. Fractions -- It is easier to add/subtract fractions than it is to split them into a sum or difference.

3. Reciprocal Functions -- Look for the side with more reciprocal functions (cotangent, cosecant, and secant) because all of these can be written in terms of cosine and sine. The tangent function can also be written in terms of sine and cosine, so choose the side with the most reciprocal functions or the one with a tangent function.

4. Binomials -- Remember that binomials can be FOILed. It is easier to FOIL two binomials than to factor a trinomial when dealing with trig functions.The basic method is quite simple. Choose a side and simplify it until it looks like the other side.

Sounds easy, right? It is, if you know your identities and are patient with yourself. Don't ever be afraid to start over. If things just aren't working out, erase or throw away that paper and try again. Maybe start with the other side this time and see how that goes. If you ever come to a point where you have multiple paths you could take, try each path until you find the one that works. I think students give up too quickly on these problems--often, I can see that they were only a step or two from the answer, but didn't know their identities well enough to see it.

I can't stress enough that knowing the trig identities is most important. When you've studied and practiced enough, relationships between identities will start to jump out at you while working these problems and will make solving them much easier.
Let me show you how I would work a few problems.


In this example, I chose to work on the left side because it was more complex, contained a 'secondary' function (cotangent), and contained a fraction. If you work from the right side, it would difficult to know how to proceed because there are so many possible directions. In the first step, I rewrote cotangent in terms of sine and cosine, which should always be your first step. In the second step, I had two fractions. Since they already had a common denominator, it was simple to combine (add/subtract) them. If they hadn't had a common denominator, I would have obtained one before combining.

Let's look at another example.


In this example I chose again to work on the left side, but either would have been possible. Mostly I chose it because I prefer simplifying fractions to splitting them, as would have been necessary if I had begun with the right side. Again, my first step was to rewrite in terms of sine and cosine, then simplify. Here, I needed to obtain a common denominator before subtracting. Never forget to do this—you cannot subtract/add fractions unless they have the SAME denominator. If they are not the same initially, you have to get them to be the same. Next, I factored again and then used the Pythagorean identity.

Notice that the second to last and last steps could have been done in a different order. If I had, I may have obtained sine first, then tangent, but functions multiplied can be re-ordered, so I still would have obtained the final product.

My last piece of advice for you is to always keep your eye on your goal. It can be easy to wander off onto a rabbit trail and completely lose sight of the desired end product. Always check with the other side of the problem and look for patterns in your work that could get you there.