The Pythagorean Identities – You Only Need to Memorize ONE

Trigonometry can be so overwhelming once you start getting into the identities! If you are a good student, you quickly realize that to be any good at trig, you need to memorize a ton of stuff!

Ok this is only SORT OF true. I really enjoy trig and can do all kinds of trig problems without thinking, yet, I do not have everything memorized exactly. Instead, I have picked up little tricks along to way to minimize the amount of memorization I have to do. It becomes such second nature, that it is just like having memorized everything.

Today, I will talk about how to do this with the pythagoream identities. My trick? You only have to memorize one of them. The rest just follow.

\sin^{2}x+\cos^{2}x=1

You need to know this identity COLD – no thinking – nothing. You just know it. After this, you can use the definition of the trig functions to recall the rest. Now what happens if you divide everything by \sin^{2}x?

\dfrac{\sin^{2}x}{\sin^2x}+\dfrac{\cos^{2}x}{\sin^2x}=\dfrac{1}{\sin^2x}

Well if you know your reciprocal identities (yes you do need these – but you will use them ALL THE TIME! I use them in my trick to remembering this huge table of trig values), this is:

1+\cot^2x=\csc^2x

What if you took that original identity and divided by \cos^2x instead?

\dfrac{\sin^{2}x}{\cos^2x}+\dfrac{\cos^{2}x}{\cos^2x}=\dfrac{1}{\cos^2x}

Well thats just \tan^2x+1=\sec^2x. If you know your reciprocal identities well enough, you can do this in your head and its just as good as having them memorized. Even if not, it takes only a second to write them down and the more you see them, the more second nature it becomes. Remember – it is all about minimizing the brain time you spend on memorizing and letting it do its job of problem solving!