Hey, everyone. Now that we know how to simplify trigonometric expressions, you're going to come across a new type of problem in which you're given a trigonometric equation and asked to verify the identity. This sounds like it could be complicated, but here I'm going to show you that verifying an identity comes right back down to simplifying, just with the specific goal of both sides of our equation being equal to each other. So here, we're going to keep using our simplifying strategies, just in a slightly different context. And I'm going to walk you through exactly how to do that here.
So let's go ahead and get started. In working with these problems, you may only have to simplify one side of your equation or you may have to simplify both sides. This will become more apparent as you begin to work through a problem. So let's go ahead and jump right into our first example here where we're asked to, of course, verify the identity. Now, something that I do want to mention is that sometimes these problems may ask you to prove the identity or establish the identity but these all mean the same thing.
Now the identity that we're asked to verify here is sinθ∙cosθ1-cosθ2 and this is equal to 1tanθ. How do we start here? Well, in working with these problems, we always want to start by simplifying our more complicated side first. So in looking at this equation, it's clear that this left side is more complicated than the right side. So we're going to start by applying our simplifying strategies to that left side.
Looking at my left side, sinθ∙cosθ1-cosθ2, remember, one of our most important strategies for simplifying is to constantly be scanning for identities. This 1-cosθ2 looks familiar. Using the Pythagorean identity sinθ2+cosθ2=1, we can replace the denominator with sinθ2.
Keeping the numerator the same as sinθ∙cosθ, some canceling allows us to have cosθsinθ in the numerator. Recognizing that cosθsinθ is the cotangent of theta, which is equivalent to 1tanθ on the right side, we conclude that both sides are equal.
We have successfully verified this identity by showing that the left side of our equation is equal to the right side. Now, even though this is called verifying the identity, this is not a new identity that you have to learn. It's all about showing that two sides of an equation are equal.
So let's go ahead and take a look at our second example here. We're still verifying the identity but the trigonometric equation that we're given here is secθ2-tanθ2cos(-θ)+1 is equal to 1-cosθsinθ2.
Start with the more complicated side, simplify using the identity tanθ2+1=secθ2 to replace all in numerator with 1. Use the even-odd identity to simplify cos(-θ) to cos(θ). Further mathematical manipulations give both sides simplifying to fa...