 Hi and welcome to the session, let us proceed on with the solution. It says, proof that sin 3x plus sin x multiplied by sin x plus cos 3x minus cos x multiplied by cos x is equal to 0. Before proceeding on, we should be well versed with the identities that will be helping us in the question. It says cos a cos p minus sin a sin v can be written as cos a plus p, whereas cos a cos p plus sin a sin v can be written as cos a minus p. The knowledge of these identities are the key ideas that going to help us in our proof. Let us start with our proof. The left hand side given to us is sin 3x plus sin x both multiplied by sin x plus cos 3x minus cos x multiplied by cos x. Now on multiplying sin x by sin 3x and then with sin x, we have sin 3x sin x plus sin x sin x plus cos 3x cos x minus cos x cos x. Now using the formula that is mentioned above, we can write for the first circumstance that is given to us, now here if we take this 3x as our a and x as our b, similarly 3x as our a and x as our b. Here we have sin a sin v plus cos a cos b and it can be written as cos a that is 3x minus x. Right, similarly here we have cos a cos b negative and sin a sin b. So if we take negative sign common it can be reduced to cos x cos x minus sin x sin x and that will be written as cos x plus x. Also this can be written as cos 3x minus x as 2x minus cos x plus x as 2x and therefore they will get cancelled out and we are left with answer 0 that is our RHS. So LHS is equal to RHS and hence we have proved the given statement by using these two identities. So I hope you enjoyed this session and do remember your identities well before proceeding on with the solution. Bye for now.