How To Add Fractions Proof

let's do someexamples simplifying trigonometric expressions. so let's say that i have 1minus sine squared theta, and this whole thing timescosine squared theta. so how could i simplify this?

How To Add Fractions Proof, well the one thingthat we do know-- and this is the mostfundamental trig identity, this comes straight outof the unit circle-- is that cosine squaredtheta plus sine squared

theta is equal to 1. and then, if we subtract sinesquared theta from both sides, we get cosine squaredtheta is equal to 1 minus sine squared theta. so we have two options. we could either replace this1 minus sine squared theta with the cosinesquared theta, or we could replace thiscosine squared theta with the 1 minussine squared theta.

well i'd prefer to dothe former because this is a more complicatedexpression. so if i can replace this withthe cosine squared theta, then i think i'msimplifying this. so let's see. this will be cosinesquared theta times another cosine squared theta. and so all of this is going tosimplify to cosine theta times cosine theta times cosinetheta times cosine theta, well,

that's just going to becosine to the fourth of theta. let's do another example. let's say that we have sinesquared theta, all of that over 1 minus sine squared theta. what is this goingto be equal to? well we already knowthat 1 minus sine squared theta is the same thingas cosine squared theta. so it's going to be sinesquared theta over-- this thing is the same thing ascosine squared theta,

we just saw that-- overcosine squared theta, which is going to be equalto-- you could view this as sine theta over cosinetheta whole quantity squared. well what's sine over cosine? that's tangent. so this is equal totangent squared theta. let's do one more example. let's say that we have cosinesquared theta plus 1 minus-- actually, let'smake it this way--

plus 1 plus sine squared theta. what is this going to be? well you might be tempted,especially with the way i wrote the colors,to think, hey, is there some identity for1 plus sine squared theta? but this is reallyall about rearranging it to realize that, gee, bythe unit circle definition, i know what cosine squared thetaplus sine squared theta is. cosine squared thetaplus sine squared

theta, for any given theta,is going to be equal to 1. so this is going to beequal to 1 plus this 1 right over here, which is equal to 2.