 Hi, and how are you all today? Let us discuss the following question. It says proof that cos x plus cos y the whole square plus sin x minus sin y the whole square is equal to 4 cos square x plus y by 2 But before proceeding on with the proof, we should be well versed with the identities that we are going to use As you remember a plus b the whole square is equal to a square plus b square plus 2 a b Whereas a minus b the whole square says a square plus b square minus 2 a b And then we'll be using one of the identities that is cos a plus b is equal to cos a cos b minus sin a sin b Right with the help of these identities that will be a key idea. We will be proceeding on with our proof now We'll be writing off the left-hand side first that is cos x plus cos y the whole square plus sin x minus Sin y the whole square now here we can use the property that is a plus b the whole square and a minus b the whole square and By doing so we have cos square x plus cos square y plus 2 cos x cos y Plus now we'll be expanding a minus b the whole square and we'll be having sin square x plus sin square y minus 2 Sin x sin y now here if you carefully analyze cos square x plus cos square sin square y will give us one as we know that cos square x plus sin square x is equal to 1 and similarly Cos square now here we have cos square y so cos square y plus sin square y will also give us one and then we have here 2 cos x cos y minus 2 sin x sin y further By using the identity that we discussed in our key idea if we have cos a cos b minus sin a sin b We can write it as cos a plus b if you carefully analyze what we have here. We have Cos a cos b minus sin a sin b if we'll take two common out here We'll be having cos x cos y minus sin x Sin y which can be easily written as cos a plus B right now Taking two common We have one plus cos x plus y this can be written as According to one of the identities that says one plus cos x is equal to two cos square x by Two right so here we have our x as x plus y so in the bracket we can write down two cos square x by two on opening the brackets we have four cos square X plus y by two and this is equal to our right inside So that means left inside is equal to right inside and hence We have Proved the given statement So I hope you enjoyed the session and just carefully analyze that you need to know of all your Identities before starting off with your proof because there are many identities like this one this one and Of course the identities what that we discussed in our key idea has been used to prove a given statement So bye for now