 Hi and welcome to the session. I'm Priyanka and let us discuss the following question. It says prove that sine 7x plus sine 5x plus sine 9x plus sine 3x divided by cos 7x plus cos 5x plus cos 9x plus cos 3x is equal to tan 6x. Before proceeding on with our proof, we should be well versed with two of the identities that we'd be making use of. That says sine a plus sine b is equal to 2 sine a plus b by 2 cos a minus b by 2 whereas cos a plus cos b is equal to 2 cos a plus b by 2 cos a minus b by 2. The knowledge of these two identities are the key idea of the question that will help us in proving the above statement. Let us start with our proof of the left-hand side. Now in the left-hand side, we are given and now in this we'll be making use of the identities that we discussed in our key idea above. In the numerator, there are everything that is of sine and in the denominator we'll be using the identity of cos and on doing it, if our a is 7x and b is 5x for the first bracket and a is 9x and b is 3x for the second bracket, it will be formed as 2 sine a plus b by 2 cos a minus b by 2 whereas here 2 sine a plus b by 2 cos a minus b by 2 divided by. Now here cos a plus cos b will be written as 2 cos a plus b by 2 cos a minus b by 2 and again 2 cos. Now our a and b have changed to 9x plus 3x by 2 and cos 9x minus 3x by 2. On solving we have 2 sine 6x. Now why 6x? 7x plus 5x will give us 12x and 12x divided by 2 will be 6x. Here 7x minus 5x will be 2x and 2x divided by 2 will be given as x plus 2 sine 6x because their sum is 12 and here their difference is 6 so it will be cos 3x divided by 2 cos 6x cos x plus 2 cos 6x cos 3x. Now if you carefully analyze the part here we can take out 2 common from both the brackets as well as sine 6x whereas in the denominator we can take out 2 common and cos 6x right. So 2 sine 6x is taken out common and we are left with cos x plus cos 3x divided by 2 cos 6x and here we are left with cos x plus cos 3x again. They will get cancelled out and 2 will get cancelled out and now here we are left with sine 6x divided by cos 6x. Now what does sine x divided by cos x means? It means tan x right? Now here our x is 6x so the answer will be tan 6x that is our right inside. So left inside is equal to right inside and hence