Question 1170448

{{{(cos(2x)+cos(4x))/ (sin(2x)-sin(4x))}}}


use identities:

{{{sin(2x)= 2sin(x)cos(x)}}}
{{{cos(2x)=cos^2(x) - sin^2(x)}}}
{{{sin(4x)=4sin(x)cos^3(x) - 4sin^3(x)cos(x)}}}
{{{cos(4x)=sin^4(x) + cos^4(x) - 6sin^2(x) cos^2(x)}}}


{{{(cos(2x)+cos(4x))/ (sin(2x)-sin(4x))}}}

={{{(cos^2(x) - sin^2(x)+sin^4(x) + cos^4(x) - 6sin^2(x) cos^2(x))/(2sin(x) cos(x)-(4sin(x)cos^3(x) - 4sin^3(x)cos(x)))}}}.............use {{{sin^4(x) + cos^4(x)=2 cos^4(x) - 2 cos^2(x) + 1}}}


= {{{(cos^2(x) - (1-cos^2(x))+(2 cos^4(x) - 2 cos^2(x) + 1) - 6(1-cos^2(x)) cos^2(x))/(2sin(x) cos(x)(1-(2cos^2(x) - 2sin^2(x))))}}}

= {{{(2 cos^2(x) (2 cos(2 x) - 1) )/(2sin(x) cos(x)(1-2(cos^2(x) - sin^2(x))))}}}......use {{{cos^2(x) - sin^2(x)=cos(2x)}}}


= {{{(2 cos^2(x) (2 cos(2 x) - 1) )/(2sin(x) cos(x)(1-2cos(2x)))}}}....simplify


={{{(2cos^2(x) cross((2 cos(2 x) - 1)))/(2sin(x) cos(x)(-cross((2cos(2x) - 1))))}}}


={{{(cross(2)cos^cross(2)(x) )/(-cross(2)sin(x) cross(cos(x)))}}}...simplify


={{{cos(x) /(-sin(x) )}}}


= {{{-cot(x)}}}