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On this video, we verify the following two trig identities.
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We're going to be previewing trig identities and for the one on this video, you'll need to
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know these identities, the fundamental identities and the Pythagorean identities and you should
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know these well.
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Alright, let's try to verify this identity.
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So I like to work on the sides that look a little more complicated and to me that looks
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like the left-hand side because it's in terms of tangents and sequents instead of signs
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and cosines.
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So I'm going to start on the left and put everything in terms of signs and cosines and
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see where it goes.
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So I have a negative 1 over, now tangent is sign over cosine and secant is 1 over cosine.
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Now remember you have to put the argument, you can't just write sign over cosine, you
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have to put sine of x over cosine x, don't forget that.
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Well hey that's nice, I have a common denominator so this is negative 1 over sine x minus 1 all
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over the cosine of x and then I can multiply by the reciprocal.
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So I have a negative 1 times the reciprocal of what's in this denominator, cosine x over
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sine of x minus 1.
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Now I'm going to write down where we're trying to go so we remember.
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Okay, so remember over here is what we're trying to get to.
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Now sign x minus 1, you know if that was 1 minus sine x we could multiply it times 1 plus
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sine x and then we can get 1 minus sine squared and get that cosine squared somehow in there
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so I'm kind of checking out that's where I'm going here.
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Notice I've got a negative 1 here.
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So it doesn't matter where I put that negative sign right it could be in the bottom so watch
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this, I'm going to switch that and write that as 1 over negative 1 and when you multiply
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negative 1 times this you simply get 1 minus sine x.
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So now I'm going to go ahead and multiply numerator and denominator by 1 plus sine x.
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Alright so let's see, whoops that went a little bit too far.
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Okay, so in the denominator, I see, and so in the numerator I have cosine x times 1
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plus sine x and check it out.
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Look what I'm trying to get.
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I'm trying to get a 1 plus sine x in the numerator so I'm getting closer.
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And in the denominator I have a difference of 2 squares, 1 minus sine squared x which
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ta-da, the varying identity is going to be cosine squared x so that's in the denominator
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cosine squared x, that's going to count with one of those cosine x and we're almost there.
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So the last thing is that's 1 plus sine x over cosine x and remember that's what we were
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trying to get.
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So we're done.
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So this took a little bit of writing.
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So here we go back up to beginning.
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Alright there's where we started.
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So we start with this identity and we're just going to work on the left side saying these
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are all equal until I finally get it to look like it is on the right side.
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So I started off by writing everything in terms of signs and cosines.
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Then I had a common denominator so I just simplified the denominator.
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And then I multiplied by the reciprocal.
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And then I kind of finagled this since I had a negative 1.
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Instead of thinking of the negative cosine in the numerator I wanted to rewrite the sine
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x minus 1 is 1 minus sine x and that could only be if I multiplied by negative 1 and there
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was a negative 1 there.
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So that worked out nicely.
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If not, if there was no negative 1 you could change that by multiplying the top and bottom
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by negative 1 and it would have been a different problem in that case.
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So we have then the cosine x.
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Then the next thing is I want to get a 1 plus sine x in the numerator.
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So you have to keep in mind where you're trying to get to.
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So how about if I just multiply the top by 1 plus sine x and I do the same thing in the
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denominator and lo and behold I get this but that green identity in the denominator and
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that gets us finally to our final answer.
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So again try this all on your own.
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First copy the way I've done it.
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Make sure you understand my steps.
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Think it out your own piece of paper and start from the beginning and see if you could
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do it.
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And as always this is just one way of solving this trick identity.
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There isn't just one way of doing it as long as from one step to the next.
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You are using correct algebraic steps or correct trick identities.
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Alright this actually is a pretty easy one to do.
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So how about putting the video on pause and see if you can get this one on your own.
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Okay I'm going to do it now.
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This looks fun.
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Well obviously the left side is the more complicated looking one.
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So we're going to start with that.
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And I notice I have a sine square times the cosecant square.
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I'm going to use light put things in terms of sine and cosine.
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Cosecant of x is one over sine of x.
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So cosecant square x is one over sine square x.
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These are reciprocals.
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When you multiply reciprocals you get one.
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Now so you could either just change that to one or you could actually write that out.
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That's the same thing as sine square x times one over sine square x plus the other sine
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square x all over cosine square x.
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And then the sine square is cancel.
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So I have one minus one plus sine square x over cosine square x.
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Ooh beautiful.
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So that's sine square x over cosine square x.
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Which is what we want.
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Look at we want tangent square x.
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That's exactly what tangent square x equals.
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Ta-da.
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We just showed that we can do it.
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It was not nice to do one.
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It wasn't so crazy difficult.
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So sometimes you get lucky.