Video Transcript
Which the following expressions is
equivalent to cos 12𝑥 plus sin squared six 𝑥? A) cos squared six 𝑥 minus sin
squared six 𝑥, B) two cos squared six 𝑥, C) two cos squared six 𝑥 plus one
minus sin squared six 𝑥, D) one minus sin squared six 𝑥, or E) two sin six 𝑥 cos
six 𝑥.
Well, to be able to actually show
which one of the expressions is equivalent to our expression here, what we’re gonna
need to use is one of the double angle formulae. And these are a set of formulae
that actually, yeah, you need to so to revise and remember. And here’s the one that we’re gonna
use today.
So this formula states that the cos
two 𝜃 is equal to cos squared 𝜃 minus sin squared 𝜃. To be able to use this in this
question, we’re actually gonna say that 𝜃, so the 𝜃 in this formula, is gonna be
equal to six 𝑥. So therefore, we can say that the
cos two 𝜃 would be the same as the cos 12𝑥. We look back at our original
expression. We’ve actually got a cos 12𝑥.
Now because we know it’s the same
as cos two 𝜃 in our double angle formula, we can actually express it now like this,
which is cos squared six 𝑥 minus sin squared six 𝑥, and then plus our original sin
squared six 𝑥. We can actually now simplify this
even further cause we’ve got cos squared six 𝑥 then minus sin squared six 𝑥 plus
sin squared six 𝑥. So these will actually cancel each
other out, which will give us cos squared six 𝑥.
Great! Okay, does this match any of our
answers on the left-hand side? But actually it doesn’t yet. So what we’re gonna use, we’re
gonna use one of our trigonometric identities. And this identity states that cos
squared 𝜃 plus sin squared 𝜃 equals one. And this is probably one of the
most common ones that you will use. And then we can actually rearrange
it to give us cos squared 𝜃 is equal to one minus sin squared 𝜃.
And great! Now this means this is gonna be
something we can use to actually change our cos squared six 𝑥 into a different
form, which now how when we see it will give us one minus sin squared six 𝑥, cause
we’re gonna use the trig identity to help us get that. So then we can look at our answers
on the left-hand side and see that one minus sin squared six 𝑥 is equivalent to cos
12𝑥 plus sin squared six 𝑥.
