Proof cos(A-B)=cosAcosB+sinAsinB Video Lecture - Class 12

and this is where the identities were
born who's ready to prove the cosine of
the difference of angles this guy what
am I looking at I'm looking at the
cosine of the difference of these two
angles what's it going to be it's the
cosine of a cosine of B these are the
same the signs are different the sine of
a the sine of B and then we need to get
a nice tight shot of this over okay not
that type right let's come back over
here and we're looking at our unit
circle and this is the angle P then that
point right down the unit circle is the
cosine be the sine of B all right and
then if we're looking at this a that
point right there is the cosine a the
sine of a what's the key to this is the
distance the distance between these two
points I want to perform an angular
transformation down here I'm taking that
guy where this is the difference of
angles and I'm taking this little white
line and I'm coordinate right here and
I'm claiming that those distances are
the same I just might know that and
that's why I'm at there so the claim is
these distances are the same so I'm
gonna need my distance formula here so
that's the difference of the x value
squared plus the difference of the Y
value squared and then we're going to
take the square we write there so we
need to look at these points all right
so the distance from this one here to
here this is going to be the square root
of wait for it the difference of the X's
that's the cosine a minus the cosine B e
and Dan that ding-dang is going to be
squared and we're going to subtract off
the side of it
- the sign of B whoa water my colors got
switched but I think you know what I'm
talking about this sign of B squared
that guy right there and then that's
gonna be equal to this distance right
here so can you see that behind me or
right on a bear oh I'll go back Andy
yeah yeah cuz this is going to be equal
to can you see no that's behind my head
okay I'll go forehead it now I'll go
over here so then yeah that's the square
root of the differences of the X's
that's the cosine of a minus B and that
should have been in blue minus 1 that
whole dang thing is going to be squared
and then minus the differences in the
Y's so then this one's in blue this is
the sine of a minus B uh-huh parentheses
minus 0 and then that dang tank is gonna
be squared get that root right over
there all right I'm over here and we're
gonna clean this up by doing some
algebra but first we square both sides
it gets rid of our roots and now we're
left with these guys I'm going to
multiply that out I'm using my special
products this is gonna be this is gonna
be the cosine squared a minus 2 times
the cosine a cosine B and then plus the
cosine a squared B and that's how this
here multiplies out C Wow
this is going to get tough so then this
is minus this entire quantity plus plus
this entire quantity the sine squared a
minus 2 times this sine a sine B plus
the sine of squared B and then that
is this guy expand it out alright and
then that's going to be equal to dag I
think I've run out of room so I'm going
to put that right here so then that's
not when I square that I'm gonna get the
cosine squared a minus B this one times
that one double it minus two times the
cosine of a minus B plus one now here's
the tough part and the part that people
seem to miss the sign - that is just the
sign so when I square that dang thing
it's going to be plus the sine squared a
minus B and now we're going to clean
this up because I'm looking over here on
this left hand side when I see these two
guys the cosine of a in the sine of a
and both of those are squared that's
going to be one fun and then I also see
the cosine of B and the sine of B both
of them squared that's equal to one fun
Pythagorean so then on the left side I'm
left with these guys and I'm going to go
ahead and I'm going to factor out that
minus 2 I hope I'm not stepping too many
skips remember you can rewind hey this
one is the same as that one
so no those are different but they both
have a minus two so I'm going to do
minus two times that quantity the cosine
of a the cosine is BP plus because I
factored out that minus the sine of a
sine of B now we're almost there see I'm
gonna close that up and I'm done with my
left-hand side over there on the right
hand side that's this business down here
whoa wait a minute
Pythagorean once more I see and explore
the cosine squared the sine squared of
the argument doesn't matter what the
argument is that's going to be one plus
that one then it's gonna be minus
two times the cosine of a minus B all
right now I'm going to collect like
terms okay that's two there that's two
there so if I have two on each side then
I can just rid myself of those twos lose
just think about linear equations let me
write what I have now I have minus two
times the cosine a cosine B plus sine a
sine B that's going to be equal to I
minus two times the cosine of a minus B
now stay with me the only thing I have
left to do is divide both sides by that
-2 and when I do I'm going to get that
the cosine of a cosine of B plus the
sine of a the sine of B that's going to
be the cosine of a minus B and what do
we see this guy right here where the
cosine of a minus B is that stuff it
hasn't shown