Video Transcript
Simplify cos 𝜃 times csc 90
degrees minus 𝜃 minus tan 𝜃 times tan 90 degrees minus 𝜃.
Whenever you see something like csc
90 degrees minus 𝜃 or tan 90 degrees minus 𝜃 or cos 90 degrees minus 𝜃, some
trigonometric function evaluated at 90 degrees minus 𝜃, you should think about
cofunction identities. And here they all are.
This identity tells us that we can
rewrite csc 90 degrees minus 𝜃 as sec 𝜃. So making this substitution, we get
cos 𝜃 times sec 𝜃 minus tan 𝜃 times tan 90 degrees minus 𝜃.
And this identity tells us that we
can replace tan 90 degrees minus 𝜃 by cot 𝜃, like so. So now we have cos 𝜃 times sec 𝜃
minus tan 𝜃 times cot 𝜃.
We can use the definition of the
reciprocal trigonometric functions to write this expression in terms of sin, cos,
and tan alone. For example, we can replace sec 𝜃
by one over cos 𝜃 to get cos 𝜃 times one over cos 𝜃 minus tan 𝜃 times cot
𝜃. And we can use the definition of
cot 𝜃: cot 𝜃 is equal to one over tan 𝜃.
And now we have an expression,
which is written in terms only of cos 𝜃 and tan 𝜃. We can see that there’s further
simplification to be had here. We’ve got cos 𝜃 times one over cos
𝜃, and so those cos thetas cancel out.
And similarly, in tan 𝜃 times one
over tan 𝜃, the tan thetas cancel out. This leaves us simply with one
minus one. So the answer is zero; cos 𝜃 times
csc 90 degrees minus 𝜃 minus tan 𝜃 times tan 90 degrees minus 𝜃 is equal to
zero.
