 Hi and welcome to the session. Let us discuss the following question which says, proof the following and we have cos 4x plus cos 3x plus cos 2x upon sin 4x plus sin 3x plus sin 2x is equal to cos 3x. So let us begin with the solution and we will start with the left side of the problem which is cos 4x plus cos 3x plus cos 2x upon sin 4x plus sin 3x plus sin 2x and show that it is equal to the right hand side which has got 3x. First let us learn a simple identity which is cos a plus cos b is equal to 2 cos a plus b upon 2 into cos a minus b upon 2 and second is sin a plus sin b is equal to 2 sin a plus b upon 2 into cos a minus b upon 2. So these are identity 20 of your book and with the help of these identities LHS can further be written as now LHS of this first identity is cos a plus cos b. So on taking cos 4x and cos 2x and applying this identity first let us rearrange. So we have cos 4x plus cos 2x plus cos 3x and in the denominator also sin 4x plus sin 2x plus sin 3x which is equal to cos a plus cos b and here in place of a we have 4x and in place of b we have 2x can be written as 2 cos 4x plus 2x is 6x upon 2 into cos 4x minus 2x is 2x upon 2 plus cos 3x and in the denominator sin a plus sin b where a is 4x and b is 2x applying this identity we have 2 sin a plus b that is 4x plus 2x which is 6x upon 2 into cos a minus b. So 6x minus 2x is 2x upon 2 plus sin 3x which is equal to 2 cos 3x into cos x plus cos 3x upon 2 sin 3x into cos x plus sin 3x. Now from the numerator taking cos 3x common we have 2 cos x plus 1 and from the denominator taking sin 3x common we have 2 cos x plus 1. Now 2 cos x plus 1 is common in the numerator and denominator so in cancelling we have cos 3x upon sin 3x and cos upon sin is cot so we have cot 3x which is equal to the right hand side and hence we have cos 4x plus cos 3x plus cos 2x upon sin 4x plus sin 3x plus sin 2x is equal to cot 3x hence proved. So this completes the solution hope you enjoyed it take care and have a good day.