Question

Find the points on the curve y=(cosx)/(2 sinx) at which the tangent is horizontal

Answer 1

$y=\dfrac{\cos x}{2\sin x}=\dfrac12\cot x\implies y'=-\dfrac12\csc^2x=-\dfrac1{2\sin^2x}$

The tangent lie is horizontal whenever $y'=0$. But this is never the case, since $\sin^2x>0$ and so $y'$ for all $x$, so horizontal tangents exist.