 Hello friends! So continuing with our journey on trigonometry, so we are today going to discuss fundamental relations between the trigonometric ratios of an angle. Now, the relations which we are going to discuss in this session and the next session probably will be very, very useful all through your trigonometric journey. So till whatever level you take up trigonometry, these basic relationships are going to be very, very useful. So paying attention to this and hence taking note of it and then using these relations into problem solving will be very, very crucial. So let us see the first and the very basic relationship. So in this case I have shown a triangle ABC which is right angle at A and let us say this side BC is hypotenuse H and this side AB is base B and this side AC is perpendicular. Now by Pythagoras theorem, by Pythagoras theorem what do we know? By Pythagoras theorem we know that H square is equal to H square is equal to P square plus B square and also let us assume this angle ABC is nothing but theta. Now we know that H square is equal to P square by B square. So hence now dividing this equation by H square both sides I can divide by H square because H is not equal to 0 clearly H is not equal to 0 because it's a hypotenuse of a right angle triangle with some definite area. So hence H is not equal to 0 so division by H square is allowed. So hence I can say H square upon H square is equal to P square by H square plus B square by H square, isn't it? Now what will H square by H square be? So this is nothing but 1 and now let us focus on P by H. So let me write this as P upon H whole square. I can take the power 2 as common and then here B upon H whole square, isn't it? Now let us just understand what is P by H. Now if you look closely into this triangle then P upon H is nothing but opposite by hypotenuse if you see opposite by hypotenuse to theta and this is nothing but sin of theta, isn't it? Similarly B upon H is equal to adjacent upon hypotenuse and this is nothing but cos theta, right? So hence I can replace this relationship as and I can take LHS or I reverse these two terms. So I take RHS as LHS now or rather let me write it for avoiding any confusion. So P by H square plus B by H whole square can be written as 1. I just swapped the two sides. Now what is P by H? We just found out this is equal to sin theta whole square plus cos theta whole square and this is equal to 1 which is now written as sin square theta. So it is sin theta whole square can also be written as sin square theta plus cos square theta is equal to 1. This is very, very famous relationship, very, very famous relationship. This can be further reduced to sin square theta. These forms should also be remembered. Sin square theta can be written as 1 minus cos square theta and similarly cos square theta can be written as 1 minus sin square theta, isn't it? Now same can be further simplified. So 1 minus cos square theta can be written as 1 minus cos theta times 1 plus cos theta, isn't it? Because it is A minus B form. So I am writing it here. It is A square minus B square form which is nothing but A minus B times A plus B. So you can use it here. So it is again 1 square. If you look closely this is also 1 square minus sin square theta. So hence it is 1 minus sin theta times 1 plus sin theta. This is how. So you remember this form, very, very important and these two forms as well. In the next session we will take up the next relationship.