 Hi, I'm Zor. Welcome to a new Zor education. This is the second lecture in the category of simple identities. We have already proven a couple of simple identities in the previous lecture. I'll just continue talking about these identities, and they are as simple, just need a little explanation maybe that's it. So, let me just start one by one. Sine of pi over 2 plus pi equals to cosine of pi. Well, let me put here something which we have actually learned before on the previous lecture. Sine of pi is equal to cosine of pi pi over 2 minus pi, and cosine of pi is equal to sine of pi over 2 minus pi. Very obvious in the case of acute angles and right angles, because these two angles complement each other to 90 degree. Now, this is a slightly different thing. However, however, we did learn some other properties of the sine, and that's exactly what I'm going to apply here. Well, you can call it a trick or something which just came to my mind. Why did I do it? Well, I don't know, but that's exactly the purpose of going through all these problems to introduce you to certain tricks if you wish. I know that sine of pi minus pi is exactly the same as sine of pi. Remember why? Because if this is pi and this is pi, then this is pi minus pi. Pi minus pi. So this point and this point, they are symmetrical in their ordinance, which are sine. Sine is an ordinance of the point, the y coordinate if you wish. Obviously, they're the same. They project to the same point here. This is the ordinance. This is the sine. So I'll use this property. So instead of this, I can write sine of pi minus pi over 2 minus pi. That's the same thing. If this is an angle, this is pi minus angle. And these are exactly the same. Now, this is sine of pi minus pi over 2 minus pi. And using one of the equations which we have already proven before, I can see that this is equal to cosine of that. That's it. So a little trick was to remember that pi minus pi has exactly the same sine as a pi. And I substitute instead of this angle, pi minus this angle, which is this, without changing the value of the function. And then everything actually follows. And I'll use the same trick everywhere for all these little identities which I have to prove. Cosine of pi over 2 plus pi is equal to minus sine of pi. Now, again, remember what is a cosine. Cosine is abscissa. And these two, which is pi and pi minus pi, their abscissa are equal in absolute layer and opposite in sines. So cosine of pi minus pi is equal to minus cosine of pi. So that's what I'm going to use. And I will use it exactly here. So instead of this, I will put minus. This is the minus. Cosine of pi minus this, which is pi minus pi over 2 minus pi equals minus cosine of pi over 2 minus pi. Now, cosine of pi over 2 minus pi is equal to sine. Now, we have a minus sign. And that's what it is. That's equal. So I'm using exactly the same trick. And I'll use it in other cases as well. Other cases are tangent of pi over 2 plus pi is equal to minus cotangent pi. Well, let's do it this way. I know the tangent by definition is sine over cosine, right? So it's not a big deal, really, if I will use this, right? And replace their values with whatever I have already found before. Sine of pi over 2 would be cosine of pi. Cosine of pi over 2 plus pi would be minus sine pi. Now, what is cosine over sine? That's a cotangent and the minus sign. So that's how it is proved. Next is cotangent equals minus tangent. Exactly the same thing. What is cotangent? It's a cosine over sine by definition. Equals cosine of pi over 2 plus pi is minus sine, minus sine pi. Sine of pi over 2 is cosine. Now, what is sine over cosine? It's tangent and minus sine. Here it is, so there we go. That's the proof. Couple of more. Exactly the same way, using the definition and the previously proven identities. Secant of pi over 2 plus pi is equal to minus cos secant of pi. What's the definition of secant? 1 over cosine. Now, cosine of pi over 2 plus pi is minus sine. So it's minus sine of pi. What is 1 over sine? It's a cosecant and the minus sign. So let's equal it. And the last one is cosecant of pi over 2 plus pi equals secant of pi. Same thing. Cosecant is 1 over sine. Now, sine of pi over 2 plus pi is equal to cosine of pi. And this is the definition of a secant and the story. All right, so I do suggest you to go through the notes to this lecture at Unisor.com. And just do it yourself. It's a very simple exercise, and it will prepare you for something more difficult. That will be in the next lectures. Anyway, good luck. Thank you.