hey guys and welcome to another Tuesday trick where today we'll be looking at
cos double angle identities and picking the right one right so there's
3 different cos double angle identities which makes it quite triggy when knowing
which to pick so when it comes to trig proofs there is a simple trick
so let's kick off.
So there are three cos double angle identities and what are
they well how'd that get there, I'm just a Mathemagician these days okay so the
key is when we have trig proofs how do we know which one to pick
in order to understand which one to pick we need to understand what are the
purpose of identities and the purpose is to simplify the problem so when we have
a proof we want to use identities to simplify the problem now in maths what's
one way simplify the problem well it's reduce the number of terms so we
can reduce the number of terms in a problem that makes life a whole lot
easier for us so that's the trick we're going to use here we want to use the cos
double angle identity that reduces the number of terms so how can you recognize
that trick okay look at this 1 plus cos 2 theta equals 2 cos squared 1 minus cos
2 theta equals 2 sine squared that's just by rearranging this so if you have
a double angle in the question for cos and then you've got a plus one next to
it that means we use this identity because the 1 will cancel with the minus 1
and same with here we've got a minus so that minus will cancel with that one so
the aim is for the trick with the cos I'm sorry the trick for the cos double
angle identity is pick the cos double identity that will eliminate the one
which is obviously going to be one of these two this identity is rarely ever
used in trig proofs it has other uses in other places but in trig proofs it's
usually going to be one of these two because we want to eliminate the one in there so
whiteboard's clean and let's look at an example now of how to apply this trick
let's consider this example proof 1 plus cos 2 theta plus sine 2 theta over
- cos 2 theta plus sine 2 theta equals cot theta so clearly here we can see
we've got double angles everywhere and we got this one plus 1 - so what we're
going to do is we'll start with the left hand side and straightaway we know the
trick is we want to sub the 1 plus we're subbing the 2 cos squared identity
because that cancels the 1 so that just turned into 2 cos square theta now
what's sine 2 theta well luckily sine 2 theta double angle identity there's
only one option to pick from which is plus 2 sine theta cos theta over now
we've got 1 minus cos 2 theta so which one cancels the 1 well that's
the sin squared identity and you should know 1 minus cos 2 theta is just 2
sine squared theta it and again sin 2 theta only has one option so from this
point to help us I did a tutorial a couple weeks ago on the manipulation
this for trig proofs which I'll leave a link up to here so the manipulation list
is E F a /s s so at this point what's my next step, do I need another identity?
well let's go through the list do I have anything to expand
no no expansions factorize yes all right lots of common factors here so we can
factorize by taking 2 cos theta out and we're left and then on the bottom take
to sign feet are out and we're left with sine theta plus cos theta and then look
at this they cancel the twos cancel I'm left with cos theta over sine theta
we're trying to prove that equals cot theta go down a list that means we need
another identity but that is just cot theta by definition which equals the
right hand side so you can see by knowing this trick with the cos double angle
identity it makes solving these proofs really easy so the key is to know which
one it is and it's the one that eliminates the one, sin double angle identity is
just 2 sine theta cos theta so there's no problem with sin 2 theta it's just the
cos double angle identities so I hope you enjoyed this tutorial and I'll see
you tomorrow for Wednesday.
Whoa you made it to the end thank you
for watching and I hope you learned something awesome today
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