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CCSS.Math:

Let's do some
examples simplifying trigonometric expressions. So let's say that I have 1
minus sine squared theta, and this whole thing times
cosine squared theta. So how could I simplify this? Well the one thing
that we do know-- and this is the most
fundamental trig identity, this comes straight out
of the unit circle-- is that cosine squared
theta plus sine squared theta is equal to 1. And then, if we subtract sine
squared theta from both sides, we get cosine squared
theta is equal to 1 minus sine squared theta. So we have two options. We could either replace this
1 minus sine squared theta with the cosine
squared theta, or we could replace this
cosine squared theta with the 1 minus
sine squared theta. Well I'd prefer to do
the former because this is a more complicated
expression. So if I can replace this with
the cosine squared theta, then I think I'm
simplifying this. So let's see. This will be cosine
squared theta times another cosine squared theta. And so all of this is going to
simplify to cosine theta times cosine theta times cosine
theta times cosine theta, well, that's just going to be
cosine to the fourth of theta. Let's do another example. Let's say that we have sine
squared theta, all of that over 1 minus sine squared theta. What is this going
to be equal to? Well we already know
that 1 minus sine squared theta is the same thing
as cosine squared theta. So it's going to be sine
squared theta over-- this thing is the same thing as
cosine squared theta, we just saw that-- over
cosine squared theta, which is going to be equal
to-- you could view this as sine theta over cosine
theta whole quantity squared. Well what's sine over cosine? That's tangent. So this is equal to
tangent squared theta. Let's do one more example. Let's say that we have cosine
squared theta plus 1 minus-- actually, let's
make 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 for
1 plus sine squared theta? But this is really
all about rearranging it to realize that, gee, by
the unit circle definition, I know what cosine squared theta
plus sine squared theta is. Cosine squared theta
plus sine squared theta, for any given theta,
is going to be equal to 1. So this is going to be
equal to 1 plus this 1 right over here, which is equal to 2.