### Video Transcript

Consider the identity sin squared
π plus cos squared π equals one. We can use this to derive two new
identities. First, divide both sides of the
identity by sin squared π to find an identity in terms of cot π and cosec π.

Then thereβs a second part to the
question that weβll come on to in a bit. So, in this question, the first
thing weβre asked to do is divide our identity by sin squared π. And when we do that, we get sin
squared π over sin squared π plus cos squared π over sin squared π is equal to
one over sin squared π. Well, the first term is the easiest
to deal with, because of weβve got sin squared π divided by sin squared π, weβre
just going to have one.

Well, if we take a look at the
second term, if weβve got cos squared π over sin squared π, well, we can use one
identity to help us here. And that is that sin π over cos
π, or similarly sin squared π over cos squared π, is equal to tan π, or tan
squared π. But as we can see, this is the
other way up to our second term. So, what do we do here?

Well, if we rearrange, we get that
cos π over sin π is equal to one over tan π. So, great! But is this useful? Well, not quite yet, because we
want our identity in terms of cot π and cosec π. And this is neither of these
yet. However, cot π is equal to one
over tan π. So therefore, cot squared π will
be equal to one over tan squared π. So, that means we have our second
term. Great! So, now all we need to do is move
on to our last term.

And now to find this term in terms
of cot and cosec, we can use one of our other identities. And that is that cosec π is equal
to one over sin π. So therefore, similarly cosec
squared π is gonna be equal to one over sin squared π. So therefore, weβve created one of
our new identities. And that one is that one plus cot
squared π is equal to cosec squared π. Okay, great! So, now letβs move on to the second
part of the question.

So, now for the second part of the
question, we need to divide both sides of the identity through by cos squared π to
find an identity in terms of tan π and sec π.

So, when we do that, we get sin
squared π over cos squared π plus cos squared π over cos squared π equals one
over cos square π. So, once again, weβll start with
the easiest term. And that is cos squared π over cos
squared π is just gonna be equal to one, because anything divided by itself is just
one. So, for the first term, weβre gonna
look at one of the identities we looked at earlier, because weβve got sin squared π
over cos squared π.

Well, we know that sin π over cos
π is equal to tan π. So therefore, sin squared π over
cos squared π is gonna be equal to tan squared π. Well, finally, for the last term,
we have one over cos squared π. But we know that sec π is equal to
one over cos π. So therefore, one over cos squared
π is gonna be equal to sec squared π.

So, now if we quickly double check,
we can see that our identity is in terms of tan π and sec π. So, we can say that if weβve got
the identity sin squared π plus cos squared π equals one, and we divide both sides
through by cos squared π, we create the identity which is tan squared π plus one
is equal to sec squared π.