Question

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

Answer 1

`y=(\cos x)/(2\sin x)=\frac12\cot x\implies y'=-\frac12\csc^2x=-\frac1{2\sin^2x}`

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