I never knew this when I was a student but becoming a teacher this became a game Cher for me to be able to help my students remember all of the Pythagorean identities so to remember all of the Pythagorean identities it first important to understand where how do we start with one Pythagorean identity so in this case what I have is the unit circle the important thing about the unit circle here is that the radius is going to be one and we're going to have a point here on the unit circle now before going through any kind of values on the unit circle let's just say we have a random Point well we know that we can represent that here as X and as Y and we're going to have a right triangle what's important about this is that is going to be our X that is going to be our Y and this is going to represent our angle Theta so we know in this case we have the Pythagorean relationship x^2 + y^2 is equal to A1 but also one thing we talk about when we're dealing with points that are on the unit circle when this radius is one we can also represent this coordinate point as cosine of theta s of theta so by replacing my S and my cosine with Y and X I now get my first Pythagorean relationship okay so that was the first one I remember and I remember when I first opened up my textbook and I had to teach pythagorean identities I'm like oh I remember that one that one's like everybody remember sin square plus cosine Square = 1 cosine square plus sin Square = 1 like it was just ingrained in my memory however if you've ever taken a trick test or you've learn you know this is not your only Pythagorean identity and remember in the other ones unless you're a m teacher unless you're doing a ton of extra examples they can be sometimes confusing to like oh crap how do I remember these and when you have a test and you have stress and you will maybe you're a little bit confused you might be worried that you might get them wrong because a lot of times they can look very very similar and you might think oh I can use Pagan identity but it might be a plus or it might be a minus and it'll mess it all up so here is a quick easy way to be able to identify the other Pythagorean identities from this one all we simply need to do is do two steps okay the first step is oh sorry not two steps but all you need to do to find the other two is just divide by S and cosine so let's go ahead and divide everything by cosine first right because if you're going to keep a an equation similar whatever you do on the left hand side you have to do on the right hand side and when you have Expressions separated by addition or subtraction you got to make sure you divide by both of them so I'm going to divide every single term here by cosine sorry cosine squared okay so cosine square of thid cosine square of theta is going to equal 1 sin sare Theta / cosine s of theta is going to equal a tangent sareet and 1 cosine Square thet is going to equal a secant sare of thet ah that is my other one now let's go and look for one more so let's go ahead and rewrite the Pythagorean Iden again so cosine squ of theta plus sin squ of theta is equal to one now instead of dividing everything by cosine let's go and divide everything by a s Square Theta okay so cosine square of theta divid sin square of theta is going to be a c square of theta Cent a sin square of theta divid sin Square thet is going to be a 1 and a 1id sin square of thet is going to be c squared or the cosecant squared I don't know no I messed it up cosecant squared of theta and there you go ladies and gentlemen we have one two 3