 Hi and how are you all today? I am Priyanka and let us discuss the following question. It says prove that sign 3x plus sign 2x minus sign x is equal to 4 sign x cos x by 2 cos 3x by 2 But before proceeding on with the solution, we should be well versed with the identity that will be helping us To do the following question. It says that sign a minus sign b is equal to 2 cos a plus b by 2 Sign a minus b by 2 also sign 2x is equal to 2 sign x cos x once one more says cos a plus cos b is equal to 2 cos a plus b by 2 cos a minus b by 2 So the knowledge of these three identities is the key idea that is going to help us in the following proof So should we start with the proof? We'll be starting with the left-hand side Now the expression which is given to us in the left-hand side is sign 3x plus sign 2x minus sign x right now for these two or rather we'll be pairing sign 3x and sign x together and We'll make use of the identity that we have told you above that is sign a minus sign b So it will be sign 3x minus sign x in one bracket plus Sign 2x Right now here. We'll be applying the formula and on doing so we have 2 cos a plus b by 2 sign a minus b by 2 and then we'll be having sign 2x also on simplifying We have 2 cos this is 4x 4x by 2 is 2x sign 2x by 2 is x plus sign 2x right So also now here sign 2x can be written as 2 sign x cos x as we have discussed above So we have 2 cos 2x sign x plus 2 sign x Cos x now if you carefully analyze 2 sign x can be taken out as the common Element and we are left with cos 2x plus Cos x now here we can use the formula cos a plus cos b and on doing so we have 2 sign 2 cos a plus b by 2 cos a minus b by 2 where our a is 2x and b is x on simplifying it further We have 2 cos 3x by 2 cos x by 2 and on opening the brackets We have 4 sign x cos x by 2 and Cos 3x by 2 right and this was our Right inside itself so we can say that LHS is equal to RHS and hence We have proved the given statement to us. I hope you enjoyed the session bye for now