 Hi and welcome to the session, I am Asha and I am going to help you with the following question which says, prove the following, cos square 2x minus cos square 6x is equal to sin 4x into sin 8x. So let us begin with the solution and you will solve the left inside of the problem which is cos square 2x minus cos square 6x and show that it is equal to sin 4x into sin 8x. Now this is in the form of a square minus b square whose formula is a minus b into a plus b. So it can further be written as cos 2x minus cos 6x into cos 2x plus cos 6x. Now first let us learn two identities. First is cos a minus cos b is equal to minus 2 sin a plus b upon 2 into sin a minus b upon 2 and the second one is cos a plus cos b is equal to 2 cos a plus b upon 2 into cos a minus b upon 2. With the help of these two identities this can further be written as cos 2x minus cos 6x is in the form of cos a minus cos b where a is 2x and b is 6x. So in the first graph appear minus 2 sin a plus b means 2x plus 6x which is 8x upon 2 into sin a minus b that is 2x minus 6x which is minus 4x upon 2. Now cos 2x plus cos 6x can be written as 2 cos a plus b is cos of 2x plus 6x that is 8x upon 2 into cos a minus b. A is 2x and b is 6x so cos minus 4x upon 2 which is further equal to minus 2 sin 4x into sin minus 2x the next bracket to cos 4x into cos minus 2x. Now since sin of minus a is equal to minus sin a and cos of minus a is equal to cos a so it can further be written as minus 2 sin 4x into minus sin 2x 2 cos 4x into cos 2x. Since cos of minus a is equal to minus plus cos a therefore we have cos 2x and here we have sin of minus 2x which is minus of sin 2x. Now this is further equal to minus into minus is plus so 2 times of sin 4x into sin 2x and 2 cos 4x into cos 2x. Now let us rearrange these terms so 2 sin 4x into cos 4x and 2 sin 2x into cos 2x. Now we know that into cos a is equal to sin 2a so applying this identity this step can further written as sin 2 times of 4x into sin 2 times of 2x which becomes sin 8x into sin 4x or sin 4x into sin 8x which is the right hand side of the given problem and hence we can say that cos square 2x minus cos square 6x is equal to sin 4x into sin 8x hence proved. So this completes the solution hope you enjoyed it take care have a good day.