 Hello and welcome to the session. In this session, we will discuss the question which says that with the following identities, first part is cosecant of 3 over 2 plus x is equal to minus secant x and second part is cos of x plus y is equal to 1 minus term x term y over term secant x secant y. Now before starting the solution of this question, we should know some results. First is sin of alpha plus beta is equal to sin alpha cos beta plus cos alpha sin beta and second result is cos of alpha plus beta is equal to cos alpha cos beta minus sin beta. Now these results will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now in the first part we have to prove that cosecant of 3 over 2 plus x is equal to minus secant x. Now let us start with the left hand side. The left hand side is cosecant of 3 over 2 plus x. Now we know that cosecant of x is equal to 1 upon sin x. So here cosecant of 3 over 2 plus x will be equal to 1 upon sin of 3 over 2 plus x. This is equal to... Now using this result which is given as a key idea, now here let 3 over 2 be alpha and x be beta and we will apply the formula of sin of alpha plus beta. So this is equal to 1 upon sin alpha that is 3 power by 2 cos beta that is x alpha that is 3 sin beta that is x. This is equal to 1 upon sin of 3 power by 2 into cos x. Cos of 3 power by 2 into sin x of 3 power by 2 can be written as sin of pi plus pi by 2. And we write sin of pi plus theta is equal to minus sin theta. So this is equal to minus sin of pi by 2, 2 is 1. So this is equal to minus 1 therefore 2 is equal to minus 1. We can find cos of 3 power by 2 and this is equal to 0. Now putting this is equal to 1 0 into sin of 0 into sin x is 0. And this is equal to minus 1 upon... We know that 1 upon cos x is equal to... This is equal to minus sin x of the given equation. And is equal to right hand side we have wrote the identity. But with the second part, with the second part we have to prove that pi is equal to 1 minus tan x tan y. Well upon, now let us start with the left hand side. Now using the second result which is given in the key idea. Cos of x plus y will be equal to cos x cos y the right hand side. The numerator is in taking form. The denominator is in secant form. The numerator and denominator of this section by the product cos x cos phi. Dividing the numerator and denominator by the product cos x cos y. We get in the numerator we have cos x cos y cos x cos y. Now this is equal to... This is equal to 1 minus upon cos y. We can write this as 1 into 1 cos x cos y. Now here you can see that 1 upon cos x will be equal to 1 minus secant x. This is equal to the right hand side is equal to right hand side. We have proved the given identity of the given question. That's all for this session. Hope you all have enjoyed this session.