 Another tip you can use for proving trigonometric identities is the following. Try using a Pythagorean identity when squares or square roots are involved in the identity, right? So you see this identity, we wanna prove. Sine plus cosine squared is equal to one plus two sine theta, cosine theta, okay? Now the left-hand side does involve a square, so not initially, but maybe at some point a Pythagorean identity would be useful. So let's start with the left-hand side, which is sine theta plus cosine theta squared. And so without a better option, maybe we'd start off as we just foil it out. So because you're squaring this thing, this is sine theta plus cosine theta times sine theta plus cosine theta, for which when you foil that out, right? First outside, inside last, right? So you have all of those. So there's the F-O-I-L right here. You get sine times sine, which is a sine squared, theta. You're going to get a sine times cosine. Next, you're gonna get a cosine times sine. And then lastly, you got a cosine times cosine, so cosine squared, like so. And so now my trigonometric identity my sense is tingling right now, right? I have this sine squared and a cosine squared. If I put those together, this is where the Pythagorean identity is going to come into play. Sine squared plus cosine squared theta, right? You have that. And then also notice with the sine cosine and the cosine sine, all because multiplication commutes, sine cosine is the same thing as cosine sine. Those things are the same thing. So let's take that here, sine theta, cosine theta. And then for the second one, we're gonna switch it around. Sine theta, cosine theta, like so. For which then if we use the Pythagorean identity, sine squared plus cosine squared is equal to one, right? Remember that one. Cosine squared theta plus sine squared theta is equal to one. So we can take the sine squared plus cosine squared and we can switch it to a one. And then you have cosine plus sine cosine that doubles up to give you two, sine theta, cosine theta. Which then we have noticed that this is in fact the right-hand side of our identity. And therefore we've now proven the identity as a consequence of some algebraic maneuvers and the Pythagorean identity. When you have squares or square roots involved, a Pythagorean identity is probably gonna be very useful for you. And so that then gives us trigonometric identity tip number three.