﻿ The Pythagorean Identities - You Only Need to Memorize ONE - MathBootCamps

## 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!

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