3 Tricks for Hard Trigonometry Proofs - Introduction to Trigonometry Video Lecture - Class 10

all right now ready for some crazy
proofs these are tough ones
I might even falter a little bit but the
main thing to remember is that if you
use the tools I've talked about
reciprocals turning everything into
sines and cosines and the identities and
maybe even fraction trick here or there
you'll you know you can solve any one of
these it might take you well might take
twenty minutes might need to ask your
teacher but hey the important thing to
remember is that there's a lot of
different ways to do these and if you go
off on the wrong track as long as you
keep obeying your algebra you'll get
there eventually and you know one thing
I don't like one thing about teachers is
that they they tend to practice these
problems so you know remembering back to
my trig class the trig prof the trig
teacher would put up some crazy proof
and just do it like it's no big deal
and I think either they're using notes
or because you're in like seventh period
they've done it six other times that day
they just know it like the back of their
hand already but it doesn't really
convey what actually happens when even a
really good trig student solves these
things which is you have to sit there
poking at it for a while and you might
take some wrong turns and you might have
to sit there wondering thinking about it
whatever so hopefully I don't come off
quite as highly polished because I'm
doing that because I haven't practiced
but also because I don't you know I want
to show you guys the process for
actually solving these instead of some
magical thing that teachers do and by
the way this is for honors only these
are really hard proofs
so I've here in regular trig you should
not be here right now you're welcome to
stay though I mean I'm not going to kick
you out or anything but all right let's
see what happens so immediately I look
at this and it does not look good
everything is already in terms of sines
and cosines right so I can't use a
couple of my tricks and then there is a
cosine squared which is great the
problem is instead of one cosine squared
there's a two in front of it so that
kind of messes things up but you know
what I don't really see any other
options so I'm going to try and just
manipulate this top until it looks like
something I like better because I kind
of wish it was one minus cosine squared
right if it was one minus cosine squared
that would be an identity so what I'm
going to do is I'm actually going to
split this into two separate things so
I'm going to instead of one minus two
Coast
squared I'm going to make it one minus
cosine squared minus cosine squared now
you can see this was legal because if
you if I had given you this at first you
could just say Oh combine like terms
negative cosine squared negative cosine
squared is negative two cosine squared
and this one to do the opposite though
I'm going to turn something as neater
into something Messier just so I can get
this by itself because one minus cosine
squared if you look at your sine square
plus cosine squared equals one you can
rearrange that to get that this is sine
squared so I got sine squared minus
cosine upstair so where it upside yeah
upstairs I've got a cosine times sine
downstairs what to do I don't actually
know exactly where this is headed I just
know that anytime you can use an
identity it's a good idea you know what
the next thing I notice is that on the
right side here I have two separate
terms there's a tangent and there's a
cotangent and they're separated by a
minus sign which means they're two
separate terms term 1 and term 2 but on
the left I just have one giant fraction
which is basically one term so I wonder
if the thing I should do is split this
into two separate fractions I can take
sine squared over this is one of my
terms and the negative cosine squared
over this is my other term so I'm going
to do that we'll get sine squared over
cosine times sine this is already
looking good because it looks like a
sine will cancel and then minus cosine
squared over cosine times sine and now a
cosine is going to cancel so that's
pretty good
get rid of that cancel that one of these
cancels that one and we get oops I
canceled the wrong one on that cosine
cancels cosine so now the way I caught
that mistake was I looked at said huh
that's a 1 I looked up at what I wanted
to go when I was like that's a cotangent
the two did not match up the nice thing
about proofs is you know where you're
headed you don't have to worry about
getting the wrong answer or not you just
have to worry about actually getting
there so sine over cosine is tangent and
cosine over sine is cotangent so it
worked out now the key strategy I want
to point out here is that in
stead of making things neater we made a
Messier we took a to cosine squared and
turn it into a cosine squared and a
cosine squared
we took a cosine you know we took one
big fraction and turned into separate
fractions that's a trick that works in
calculus a lot to when you're trying to
get a tough problem Sony's trig and
these trig proofs sometimes you're
messing things up instead of making them
neater kind of like yeah the opposite of
what you've been doing in algebra all
these years all right here's another one
we've got sines and cosines already so I
can't you know transform anything um I
like the look I like the fact there's a
lot of signs of cosines that makes me
think I could use an identity the only
problem is I've got weird powers I've
got odd powers of sine and cosine but it
looks like the smart play it might be to
fact you know factor out a sign from
upstairs
saw so that'll bring this down to a sine
squared I can do the same thing
downstairs so if we do sine theta
you