 Hi and welcome to the session. Let us prove the following. Say sin x plus sin 3x upon cos x plus cos 3x is equal to tan 2x. 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 tan 2x. So left hand side is sin x plus sin 3x upon cos x plus cos 3x. Now first let us learn some simple identities which says that sin a plus sin b is equal to 2 sin a plus b upon 2 into cos a minus b upon 2 and second identity is cos a plus cos b is equal to 2 cos a plus b upon 2 into cos a minus b upon 2. So this is identity 20 of your book and by using these two identities left hand side can further be written as 2 times of sin a plus b that is x plus 3x 4x upon 2 into cos of a minus b that is cos of x minus 3x is minus 2x upon 2 and in the denominator we have cos x plus cos 3x which is in the form of cos a plus cos b where a is x and b is 3x. So in applying the identity it can be written as 2 cos first we have a plus b so x plus 3x is 4x upon 2 and here cos a must be that is x minus 3x which is minus 2x upon 2. 2 cancels out with 2 and in the numerator we have sin 2x to cos of minus x upon cos 2x and here cos minus x. Now cos minus x is called the numerator and denominator on cancel we have sin x upon cos 2x and since sin a upon cos a is equal to tan a so sin 2x upon cos 2x is equal to tan 2x since sin a upon cos a is equal to tan a and here the angle is 2x and hence we say that sin x plus sin 3x upon cos x plus cos 3x is equal to tan 2x hence proved. So this completes the decision. Hope you enjoyed it. Take care and have a good day.