 Hi and how are you all today? Let us proceed on with the question. It says prove that sin x plus sin 3x plus sin 5x plus sin 7x is equal to 4 cos x cos 2x sin 4x. Before starting off with the proof, we should be well versed with an identity which says that sin a plus sin b can be written as 2 sin a plus b divided by 2 cos a minus b divided by 2. The knowledge of this identity is the key idea that will help us in proceeding on with our proof. Wait, let us start with the left-hand side. Now we are given sin x plus sin 3x plus sin 5x plus sin 7x. Now we will be combining these two terms in brackets and then we will be applying the identity that we discussed in our key idea. That is sin a plus sin b can be written as 2 sin a plus b divided by 2 cos a minus b by 2. Now here also we can apply the same formula that is 2 sin a plus b by 2 cos a minus b by 2. On simplifying we have 2 sin. Now this is 4x divided by 2 will become 2x cos. This is minus 2x divided by 2 guilt give us minus x plus 2 sin. This will be 6x now and cos minus because we know that cos of minus x is cos x only we can write 2 sin 2x cos x because cos of minus x is cos x only plus 2 sin 6x cos x. Now we can see that cos x is common to both of them as well as the numerical 2. So on taking 2 cos x common we are left with sin 2x sin 6x. Now here again we will be applying sin a plus sin b so we have 2 sin a plus b by 2 cos a minus b by 2 and this becomes 2 sin 4x cos minus 2x on solving this whole. As we know that cos of minus x is cos x only so we can write it as cos 2x and on opening the brackets we have 4 cos x cos 2x sin 4x that is our right-hand side and hence we can see that left-hand side is equal to right-hand side and hence we have proved the statement that was given to us. So I hope you remember each and every identity that was explained to you in previous classes. Bye for now.