 Hi and welcome to the session. I am Asha and I am going to help you with the following question which says, proof the following, cot 4x into sin 5x plus sin 3x is equal to cot x into sin 5x minus sin 3x. So let us begin with the solution and we solve the left hand side of the problem and show that it is equal to the right hand side. So left hand side is cot 4x into sin 5x plus sin 3x. Now cot 4x is cos 4x upon sin 4x and sin 5x sin 3x is in the form of sin a plus sin b whose formula is equal to 2 sin a plus b upon 2 into cos a minus b upon 2. So sin 5x sin 3x can be written as 2 sin, in place of a we have 5x and in place of b we have 3x. So 8x upon 2 into cos a minus b that is 5x minus 3 is just 2x upon 2 which is further equal to cos 4x upon sin 4x into 2 times of sin 4x into cos x. Now sin 4x is common in the numerator and denominator. So in cancelling we are left with 2 cos 4x into cos x. Now in the right hand side we need cot x. So to get cot x what we do is multiply the numerator and denominator by sin x. Since cot x is equal to cos x upon sin x and so cos x upon sin x can be written as cot x we have 2 cos 4x into sin x and since 2 cos a into sin b is equal to sin a plus b minus sin of a minus b. So it can further written as cot x into sin 4x plus x which is equal to 5x minus a minus b that is 4x minus x which is sin 3x which is nothing but the right hand side of the given problem and hence we can say that cot 4x into sin 5x plus sin 3x is equal to cot x into sin 5x minus sin 3x hence proved. So this completes the session. Hope you enjoyed it. Take care and have a good day.