(5cos(x)-4)/(3-5cos(x)-(3+5sin(x))/(4+5cos(x)) or (5cos(x)-4)/(3-5sin(x)-(3+5sin(x))/(4+5cos(x))?
Let's take the second one first.
[(25cos2(x)-16)-(9-25sin2(x))]/[(3-5sin(x))(4+5cos(x))]=
(25cos2(x)+25sin2(x)-25)/(12+15cos(x)-20sin(x)-25sin(x)cos(x))=
(25-25)/(12+15cos(x)-20sin(x)-25sin(x)cos(x))=0 because the numerator evaluates to zero, irrespective of the denominator.
Therefore (5cos(x)-4)/(3-5sin(x)-(3+5sin(x))/(4+5cos(x))=0, and the given expression (5cos(x)-4)/(3-5cos(x)-(3+5sin(x))/(4+5cos(x)) is non-zero (in general).