Hey, everyone. Earlier, we used all of our fundamental trig identities in order to do what's called verifying an identity, where we were really just simplifying one or both sides of an equation with the goal of making them equal to each other. Now we know some more trig identities, our sum and difference identities, so we're going to continue to do the same thing. Nothing is changing here. We're still going to use our simplifying strategies along with our trig identities in order to make both sides of a given equation equal. So let's not waste any time here and jump right into our example.
In this example, we're asked to verify the identity using our sum and difference identities. The equation that we're given here is: cosπ2-θ=sinθ. Remember that whenever we verify identities, we want to start with our more complicated side. Looking at this equation, my left side is more complicated because that argument has a difference in it.
Remembering our strategies for simplifying trig expressions, we want to be constantly scanning for identities. Working on that left side here, I see that I have a difference in that argument, which tells me that I can use my difference formula for cosine. So, using that formula, the cosine of π2-θ is cosπ2cosθ+sinπ2sinθ. From here, we can simplify this further because we know the cosine of π2 is just 0, so this entire term will go away. But the sine of π2 is equal to 1.
So all that I'm left with here is 1 times the sine of theta. But one times the sine of theta is just the sine of theta. Looking at the right side of my equation, remembering that my goal is for both sides of this equation to be equal, I see that the other side is already the sine of theta. So here I have that sinθ=sinθ and I have successfully verified this identity.
You might recognize this as being a cofunction identity that we've used earlier in this course. So here we were able to verify that cofunction identity using our sum and difference identities. Now that we know how to continue verifying identities using the new identities that we've learned, let's get some more practice. Thanks for watching, and I'll see you in the next one.