Video: Simplifying Trigonometric Expression Using Cofunction Identities

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.