Hey, everyone. We just learned a bunch of different trig identities. And now that we know so many trig identities, we can begin to use those that we already know in order to come up with even more trig identities. Like, if we take our sum formula but for the same two angles, we end up with what's called our double angle identities. Now your first question here might be why. Why do we need even more trig identities? And the answer is that they're just going to keep allowing us to work through more and more trig problems with ease. So here I'm going to walk you through exactly what our double angle identities are and how to use them in order to continue simplifying trig expressions. So let's go ahead and get started.

Now I mentioned that these double angle identities come from using our sum formulas but for the same two angles. So if I take the sine of some angle theta plus that same angle theta, I end up with this expression here. Now these two terms are actually exactly the same now, so this simplifies to 2 times the sine of theta times the cosine of theta, giving me my double angle identity for sine. That sine of 2 theta is equal to 2sinθcosθ. Now we can do the same exact thing for cosine and tangent, and this ends up giving me that the cosine of 2 theta is equal to cosθ2-sinθ2, and the tangent of 2 theta is equal to 2tanθ1-tanθ2.

Now looking at our expression here, the cosine squared of pi over 12 minus the sine squared of pi over 12, I don't know the cosine or sine of pi over 12 off the top of my head, but I do recognize this as one of my double angle identities, and specifically my cosine double angle identity, the cosine squared of some angle minus the sine squared of some angle. So this is really just the cosine of 2 times that angle. So I can rewrite this cosine squared of pi over 12 minus the sine squared of pi over 12 as the cosine of 2 times my angle here, pi over 12. Now 2 times pi over 12 is just pi over 6, so this is really the cosine of pi over 6, a value that I know really well from my unit circle. So I can come to an answer rather easily, that this is just equal to the square root of 3 over 2 having used my double angle identity.

Now something that you may have noticed here is that we have 2 other forms of our cosine double angle formula. And these are just alternate forms that come from taking this first identity here and rewriting it using our Pythagorean identities. So these alternate forms are just going to help us in different scenarios to simplify different trig expressions. Now that we've seen all of these double angle identities, how do we know when to use them? Well, something that might be rather obvious is that whenever our argument contains 2 times some angle, we should use our double angle identities because that's the exact argument of all of these identities. Now something that might be less obvious is that whenever we recognize a part of the identity, we should go ahead and use these identities.

Now this is kind of what we did in our example here. We recognize that this cosine squared minus sine squared was this particular part of my cosine double angle formula. Now here, this was a bit more simple, but it's not always going to be exactly like that. Because remember that for our other identities, we had to recognize a bunch of different forms and rearranged versions of our identities in order to successfully simplify expressions and verify identities, and it's going to be no different with our double angle identities. So let's go ahead and work through another example here.

Here, we're asked to simplify the expression but not to evaluate. The sine of 15 degrees times the cosine of 15 degrees. Now remember that when simplifying an expression, we want to be constantly scanning for identities. And here, even though I don't know the sine or cosine of 15 degrees, I do recognize that this looks like part of my double angle formula for sine. Specifically here, I see the sine of theta times the cosine of theta. I'm just missing that 2. So let's see what we can do here. I'm going to go ahead and rewrite my sine double angle identity here that the sine of 2 theta is equal to 2 times the sine of theta times the cosine of theta. Now I only want this part of this identity. So what can I do here? Well, if I go ahead and divide both sides by 2, that 2 will cancel on that right side, leaving me with the sine of 2 theta over 2 is equal to the sine of theta times the cosine of theta, which is the exact expression that I'm working with here, just with theta being equal to 15 degrees. So with this in mind, I can use that double angle identity in order to rewrite this as the sine of 2 times that angle 15 degrees over 2. Now what is 2 times 15 degrees? It's just 30 degrees. So this is really a sine of 30 degrees over 2, and we have successfully simplified this expression, taking it from 2 different trig functions to just 1 trig function. Now we're not asked to evaluate here, but if we were, we easily could because the sine of 30 degrees is a value that I know really well from the unit circle.

Now as we continue to learn more and more identities, it can become overwhelming to figure out how to simplify a trig expression. But with each new identity that we learn, we have a greater ability to work through the problems that are asked of us. So now that we know our double angle identities, let's continue practicing with them. Thanks for watching, and let me know if you have questions.