SOLUTION: Prove that the equation is an identity {{{ (sin 3x + sin 5x) / (sin 3x - sin 5x) = - (tan 4x) / tan x }}}

Algebra ->  Trigonometry-basics -> SOLUTION: Prove that the equation is an identity {{{ (sin 3x + sin 5x) / (sin 3x - sin 5x) = - (tan 4x) / tan x }}}      Log On


   

Question 1004522: Prove that the equation is an identity
+%28sin+3x+%2B+sin+5x%29+%2F+%28sin+3x+-+sin+5x%29+=+-+%28tan+4x%29+%2F+tan+x+
Answer by ikleyn(52817) About Me  (Show Source):
You can put this solution on YOUR website!
.
Apply the formulas for addition and subtraction of trigonometric functions  (see the lesson
Addition and subtraction of trigonometric functions  in this site)  to the numerator and denominator.

Let start with the numerator.

sin(3x) + sin(5x) = 2*sin(%283x%2B5x%29%2F2)*cos(%283x-5x%29%2F2) = 2*sin(4x)*cos(-x) = 2*sin4x)*cos(x).   <-----   Recall that cos(-x) = cos(x).

Similarly for the denominator

sin(3x) - sin(5x) = 2*sin(%283x-5x%29%2F2)*cos(%283x%2B5x%29%2F2) = 2*sin(-x)*cos(4x) = -2*sin(x)*cos(4x).   <-----   Recall that sin(-x) = -sin(x).

Thus

%28sin+3x+%2B+sin+5x%29%2F%28sin+3x+-+sin+5x%29 = -%282%2Asin%284x%29%2Acos%28x%29%29%2F%282%2Asin%28x%29%2Acos%284x%29%29 = - %28sin%284x%29%2Fcos%284x%29%29%2A%28cos%28x%29%2Fsin%28x%29%29 = - %28tan%284x%29%29%2F%28tan%28x%29%29.

It is exactly what has to be proved.