 Welcome to the session, I am Asha and I am going to help you with the following question which says, prove the following, cos 3 pi upon 4 plus x minus cos 3 pi upon 4 minus x is equal to minus root over 2 sin x. So let us begin with the solution and we will solve the left hand side of this problem and show that it is equal to the right hand side which is minus root 2 sin x. So left hand side is cos 3 pi upon 4 plus x minus cos 3 pi upon 4 minus x. First let us learn some simple identities which say cos a plus b is equal to cos a into cos b minus sin a into sin b. Second is cos a minus b is equal to cos a cos b plus sin a sin b. Now on comparing the left hand side of this identity with the cos 3 pi upon 4 plus x, we find here that a is equal to 3 pi upon 4 and b is equal to x, thus cos 3 pi by 4 plus x can be written as cos 3 pi by 4 into cos x minus sin 3 pi by 4 into sin x. And on comparing cos a minus b with cos 3 pi by 4 minus x, we find here that a is equal to 3 pi by 4 and b is equal to x again. So applying the second identity cos 3 pi upon 4 minus x can be written as cos 3 pi upon 4 into cos x plus sin 3 pi upon 4 into sin x and thus the left hand side which is cos 3 pi upon 4 plus x minus cos 3 pi upon 4 minus x can be written as cos 3 pi upon 4 plus x is cos 3 pi upon 4 into cos x minus sin 3 pi upon 4 into sin x. Then we have a negative sign cos 3 pi upon 4 minus x is equal to cos 3 pi upon 4 into cos x plus sin 3 pi upon 4 into sin x which is further equal to cos 3 pi upon 4 into cos x minus sin 3 pi upon 4 into sin x. Now in the second bracket we have minus cos 3 pi upon 4 into cos x minus sin 3 pi upon 4 into sin x. Now cos 3 pi upon 4 into cos x cancels out with minus cos 3 pi upon 4 into cos x and we have sin 3 pi upon 4 into sin x 2 times. So we have minus 2 sin 3 pi upon 4 into sin x which is further equal to minus 2 sin 3 pi upon 4 can be written as pi minus pi upon 4 into sin x. Now pi minus pi upon 4 this angle lies in the second quadrant and in the second quadrant sin is a positive function. So we have minus 2 sin pi by 4 into sin x and we have minus 2 sin pi by 4 is 1 upon root 2 into sin x. In the numerator we have 2 can be written as root 2 into root 2. In the denominator we have root 2 sin x and now root 2 cancels out with root 2 and we are left with minus root 2 sin x which is the right hand side of the given question. Thus we have cos 3 pi upon 4 minus x minus cos 3 pi upon 4 minus x is equal to minus root over 2 sin x had proved. So this concludes the solution. Hope you enjoyed it. Take care and have a good day.