 Hello and welcome to the session. I am Asha and I am going to help you with the following question that says, prove that sin 5x plus sin 3x upon cos 5x plus cos 3x is equal to tan 4x. Let us begin with the solution and we shall start with the left hand side and show that it is equal to the right hand side. Now the left hand side is sin 5x plus sin 3x upon cos 5x plus cos 3x. First let us learn some simple formulas with the help of which we will solve it. First is sin a plus sin b is equal to 2 sin a plus b upon 2 into cos a minus b upon 2 and the formula of cos a plus cos b is equal to 2 cos a plus b upon 2 into cos a minus b upon 2. So the elitists can further be written as first we have sin 5x plus sin 3x and using the formula sin a plus sin b here we have a is equal to 5x and b is equal to 3x. So in applying the formula we have 2 sin a plus b that is 5x plus 3x upon 2 into cos a minus b that is 5x minus 3x upon 2 and in the denominator applying the formula cos a plus cos b we have 2 cos a plus b is 5x plus 3x upon 2 into cos 5x minus 3x upon 2 which is further equal to 2 sin 5x plus 3x is 8x and 8x upon 2 is 4x into cos 5x minus 3x is 2x and 2x upon 2 is x upon in the denominator we have 2 cos 4x into cos x which is further equal to 2 cancels out with 2 also cos x with cos x and we have sin 4x upon cos 4x which is equal to tan 4x which is the right hand side of the given problem and thus we have left hand side is equal to the right hand side or sin 5x plus sin 3x upon cos 5x plus cos 3x is equal to tan 4x hence proved. So this completes the solution take care and have a work day.