Question 1004522
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Apply the formulas for addition and subtraction of trigonometric functions  (see the lesson 

<A HREF=http://www.algebra.com/algebra/homework/Trigonometry-basics/Addition-and-subtraction-of-trigonometric-functions.lesson>Addition and subtraction of trigonometric functions</A>&nbsp; in this site)&nbsp; to the numerator and denominator.


Let start with the numerator.


sin(3x) + sin(5x) = 2*sin({{{(3x+5x)/2}}})*cos({{{(3x-5x)/2}}}) = 2*sin(4x)*cos(-x) = 2*sin4x)*cos(x). &nbsp;&nbsp;<----- &nbsp;&nbsp;Recall that cos(-x) = cos(x).


Similarly for the denominator


sin(3x) - sin(5x) = 2*sin({{{(3x-5x)/2}}})*cos({{{(3x+5x)/2}}}) = 2*sin(-x)*cos(4x) = -2*sin(x)*cos(4x). &nbsp;&nbsp;<----- &nbsp;&nbsp;Recall that sin(-x) = -sin(x).


Thus 


{{{(sin 3x + sin 5x)/(sin 3x - sin 5x)}}} = {{{-(2*sin(4x)*cos(x))/(2*sin(x)*cos(4x))}}} = - {{{(sin(4x)/cos(4x))*(cos(x)/sin(x))}}} = - {{{(tan(4x))/(tan(x))}}}.


It is exactly what has to be proved.