# 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!

## Subscribe to our Newsletter!

We are always posting new free lessons and adding more study guides, calculator guides, and problem packs.

Sign up to get occasional emails (once every couple or three weeks) letting you know what's new!