SOLUTION: show that {{{(sin(x)sin(2x)+sin(3x)sin(6x))/(sin(x)cos(2x)+sin(3x)cos(6x))=tan(5x)}}}

Algebra ->  Trigonometry-basics -> SOLUTION: show that {{{(sin(x)sin(2x)+sin(3x)sin(6x))/(sin(x)cos(2x)+sin(3x)cos(6x))=tan(5x)}}}      Log On


   

Question 883546: show that
Answer by cigany29(23) About Me  (Show Source):
You can put this solution on YOUR website!

Let's break it down first by applying our product-to-sum formulae!
° %28sin%28x%29sin%282x%29%29=%28cos%28x-2x%29-cos%28x%2B2x%29%29%2F2=%28cos%28-x%29-cos%283x%29%29%2F2
° %28sin%283x%29sin%286x%29%29=%28cos%283x-6x%29-cos%283x%2B6x%29%29%2F2=%28cos%28-3x%29-cos%289x%29%29%2F2
°%28sin%28x%29cos%282x%29%29=%28sin%28x%2B2x%29%2Bsin%28x-2x%29%29%2F2=%28sin%283x%29%2Bsin%28-x%29%29%2F2
°%28sin%283x%29cos%286x%29%29=%28sin%283x%2B6x%29%2Bsin%283x-6x%29%29%2F2=%28sin%289x%29%2Bsin%28-3x%29%29%2F2
Remember:
* cos(-x)=cos(x)
*sin(-x)=-sin(x)
SO:
°%28cos%28-x%29-cos%283x%29%29%2F2=%28cos%28x%29-cos%283x%29%29%2F2
°%28cos%28-3x%29-cos%289x%29%29%2F2=%28cos%283x%29-cos%289x%29%29%2F2
°%28sin%283x%29%2Bsin%28-x%29%29%2F2=%28sin%283x%29-sin%28x%29%29%2F2
°%28sin%289x%29%2Bsin%28-3x%29%29%2F2=%28sin%289x%29-sin%283x%29%29%2F2
Now lets put it together!

That looks kinda messy, so lets simplify that again and while we're at it, we'll get rid of one division by multiplying by the reciprocal!

%28%28cos%28x%29-cos%289x%29%29%2F2%29*%282%2F%28sin%289x%29-sin%28x%29%29%29
Cancel the 2's and we're left with:
%28cos%28x%29-cos%289x%29%29%2F%28sin%289x%29-sin%28x%29%29%29
Now we're going to apply the sum-to-product formulae!
°%28cos%28x%29-cos%289x%29%29=%28-2sin%28%28x%2B9x%29%2F2%29%2Asin%28%28x-9x%29%2F2%29%29=%28-2sin%285x%29%2Asin%28-4x%29%29 Again, sin(-x)=-sin(x) so it's = to %28-2sin%285x%29%2A%28-sin%284x%29%29%29
°%28sin%289x%29-sin%28x%29%29=%282cos%28%289x%2Bx%29%2F2%29%2Asin%28%289x-x%29%2F2%29%29=%282cos%285x%29%2Asin%284x%29%29
Now we'll put the whole thing together:
%28-2sin%285x%29%2A%28-sin%284x%29%29%29%2F%282cos%285x%29%2Asin%284x%29%29=tan%285x%29
which simplifies to:
%28sin%285x%29%29%2F%28cos%285x%29%29=tan%285x%29
and at last we have tan%285x%29=tan%285x%29