We just saw how to use our even-odd identities to rewrite expressions with negative arguments, and we specifically saw arguments with negative angle measures in them. However, you’re more often going to have to simplify expressions that just have variables, rather than angle measures. So let's take a look at that here.
Here we’re asked to use our even-odd identities in order to rewrite the expression with no negative arguments in terms of just one trig function. In our first example, we have the negative tangent of negative theta. This is already in terms of just one trig function, but we need to get rid of that negative argument there. I know that the negative tangent of negative theta is equal to negative tangent of theta, I can simply replace the tangent of negative theta using that identity. I want to make sure that I’m paying attention to everything that I need to keep here, so make sure that you’re still keeping that negative on the outside. Then the tangent of negative theta, I can replace that with negative tangent of theta. Now I see here that I have two negative signs and I know that two negatives make a positive, so this ends up just being a positive tangent of theta. And now we have successfully rewritten this with no negative arguments in terms of just one trig function. We're done with that example there.
Let's move on to our second example. Here we're asked to rewrite this expression, the sine of negative theta over the cosine of negative theta. Now I can go ahead and use my even-odd identities for sine and cosine. I know that the sine of negative theta is equal to negative sine of theta. Then the cosine of negative theta, because it's an even function, is simply equal to the cosine of theta. Now from here, we have gotten rid of those negative arguments, but it's still in terms of two trig functions. So how can we simplify this further? Well, the sine over the cosine is simply the tangent. So keeping that negative sign, this ends up just being the negative tangent of theta, and that's my final answer here.
You might have seen a different way to do this, and that's totally fine. There are always multiple ways to simplify. Then using our even-odd identity for tangent, we know that the tangent of negative theta is simply equal to negative tangent of theta, so we would end up getting that exact same answer, negative tangent of theta. It doesn't matter which way you choose to simplify this because you'll get the right answer regardless.
Now that we've seen how to do this using variables, let’s continue practicing. Thanks for watching, and I'll see you in the next one.
