BAM!!!. Mr. Tarrou Let's do two more examples of verifying identities. Ok, so we have one
over one minus sine theta plus one over one plus sine of theta equals two times secant
squared theta. Don't forget we want to simplify and clearly the left hand side of the identity
is much more complicated then the right. So let's see what we can do. We have two terms
and over here we have one. So there is a place to start. So I am going to take this and multiply
the first fraction by one plus the sine of theta on the top and bottom. And on the second
term, or the second fraction, do the exact same thing only this time multiply by one
minus sine theta. I didn't leave myself very much room to write here. Now I am going to
write this as one big fraction because I am setting up a common denominator as I go. We
are going to have one plus sine theta plus, now I am going to use parenthesis on the second
numerator not because it is really going to matter with this particular question because
it is plus and the parenthesis really are not going to do anything, but if it were minus
that would have to get distributed through the parenthesis because there is more than
one term in this second numerator. If you did not do that you would have sign errors
and your proof would fall part. So one times one minus sine of theta is just one minus
sine theta. The denominator is going to be, now a lot of times I will leave these in factored
form because a lot of times stuff cancels out, but I can look at the top and see that
it is going to be sine minus sine so my numerator is only going to be a value of two and there
is no chance of anything canceling out with the denominator so I am going to go ahead
and multiply this out. Again we do have the same denominator twice so we have one times
one is one. One times positive sine is plus sine, then one times negative sine is negative
sine. Well you have a positive sine and a negative sine they are going to cancel out
when adding. And this is a difference of squared pattern that you might remember from algebra
one, so the middle term is going to cancel out anyway. Let me get back to what I was
saying. One times one is one. One times positive sine is sine. One times negative sine is negative
sine. Those are going to cancel out. Negative sine times positive sine is negative sine
squared theta. Ok. Well sine minus sine, those are going to cancel out and we are left with
one plus one which is two. This is really good because that is what I need over here
is a two, two times something. I forgot my exponent of two. Now I have got one minus
sine squared theta. That does not look like this over here, so what can I do? Well I am
out of space so I could just stop and start working with the other side. I think I am
going to go ahead and do that. You might also recognize that uhmm if you want to continue
this on just one side, secant is one over cosine right. And one minus sine....actually
let me keep working with this. Secant is one over cosine, so secant squared is equal to
one over cosine squared. So I need a denominator that has a cosine function in it. Well, the
first pythagorean identity says that sine squared plus cosine squared is equal to one.
If I were to manipulate that so that one side said one minus sine squared theta, the other
side would say cosine squared theta. So one minus sine squared theta is going to come
out and with a substitution using the pythagorean identity, we get 2 over cosine squared theta.
We are just about done. One over cosine, well this is a 2, but if I bring the cosine function
up or if I remember that one over cosine equals secant, then I can bring the cosine squared
function up and write two times secant squared theta and BAM!!! I am done:D Alright. So,
please if you have a full sheet of paper, do your proofs so that the left and right
hand sides going down in columns. This horizontal work like this is not good when you have a
full sheet of paper and you would like to make it a little bit easier on your math teacher
to grade! Alright, so 2 secant squared does equal 2 secant squared. We have this proof
done, or this identity verified. One more example and I am going to move on to the next
video which is going to be half angle...or excuse me, sum and difference identities for
sine and cosine and tangent. But first, one more example. We have the sine of theta plus
cosine theta over cosine theta minus sine theta minus cosine theta all over sine theta
equals secant theta cosecant theta. Ok, so we are clearly again going to work with the
left hand side because it is obviously more complicated than the right. Well, we have
two terms and over here we only have one term which means that I am going to think that
umm...we are going to go ahead and find common denominators again and try and bring those
fractions together and see what happens. Well, sine and cosine don't have anything in common
so I am going to multiply this fraction on top and bottom by sine. Wrap that numerator
up in parenthesis because we are going to need to distribute. There are two terms up
there. Cosine comes over to this fraction again with parenthesis multiplying the top
and bottom. What do we get? We are going to distribute, so we have sine times sine is
sine squared. Sine times cosine is plus sine theta times cosine theta. Now look, minus
in the middle and this second fraction's numerator has two terms. If you do not properly use
your parenthesis, you are not going to carry that negative sign through and your proof
is going to fall apart. Make sure that you use your parenthesis judiciously, a lot, to
make sure you don't make any sign errors. Cosine times sine again which is going to
be sine theta times cosine theta. Then cosine theta times negative cosine squared theta.
All of that is over my common denominator that I set up of cosine theta times sine theta.
I am just going to write up one big fraction because I have set up my common denominator.
So sine theta cosine theta. Now see that minus sign is what was originally between the two
fractions and that minus sign has to get distributed through the parenthesis. When you do that,
what is going to happen? Well we are going to get sine cosine which is positive and negative
sine cosine, so those are going to cancel out. We have a negative times a negative which
is positive so our next line is going to be sine squared theta, negative times negative
is positive so plus cosine squared theta over sine theta cosine theta. Now, we are really
close to the answer. What is sine squared plus cosine squared? Go look at your formula
sheet or if it is to memory, the pythagorean identity, that is equal to one. And one over
sine theta cosine theta and my winter holiday decorations are getting in the way here. What
is one over sine? Cosecant. What is one over cosine? That is right! It is secant. So we
get secant theta from one over cosine, and cosecant of theta from one over sine. And
BOOM! There we go. We are done!!! Two more example are out of the way. I am Mr. Tarrou.
Go Do Your Homework! WHOOO!!!