Reduce the quotient $\frac{\cot A+\mathrm{cosec}A}{\cos A+1}$ to a single trigonometric ratio.
Which ratio is it?

Convert the numerator to sin and cos: $\cot A+\mathrm{cosec}A=\frac{\cos A}{\sin A}+\frac{1}{\sin A}=\frac{\cos A+1}{\sin A}$. Dividing by $\cos A+1$ cancels that whole bracket and leaves $\frac{1}{\sin A}=\mathrm{cosec}A$. Answering $\sin A$ turns the surviving fraction upside down. Check at $A=30$ deg: the quotient is $\frac{\sqrt{3}+2}{(\sqrt{3}/2)+1}=2=\mathrm{cosec}30^{\circ }$.