$\cfrac{{\sin (ax + b)}}{{\cos (cx + d)}}$

Let $y = \cfrac{{\sin (ax + b)}}{{\cos (cx + d)}}$

$\Rightarrow$ $\cfrac{{dy}}{{dx}} = \cfrac{d}{{dx}}\left( {\cfrac{{\sin (ax + d)}}{{\cos (cx + d)}}} \right)$

$= \cfrac{{\cos (cx + d)\cfrac{d}{{dx}}\sin (ax + d) - \sin (ax + b)\cfrac{d}{{dx}}\cos (cx + d)}}{{{{\cos }^2}(cx + d)}}$

$= \cfrac{{a\cos (cx + d)\cos (ax + b) + c\sin (ax + b)\sin (cx + d)}}{{{{\cos }^2}(cx + d)}}$

$= a\cos (ax + b)\sec (cx + d) + c\sin (ax + b)\tan (cx + d) \cdot \sec (cx + d)$

therefore $\frac{dy}{dx}=a\cos (ax + b)\sec (cx + d) + c\sin (ax + b)\tan (cx + d) \cdot \sec (cx + d)$