Simplify $\frac{\cos A}{\mathrm{cosec}A+1}+\frac{\cos A}{\mathrm{cosec}A-1}$.

Use the common denominator ${\mathrm{cosec}}^{2}A-1={\cot }^{2}A$. The numerator is $\cos A\left[(\mathrm{cosec}A-1)+(\mathrm{cosec}A+1)\right]=2\cos A\mathrm{cosec}A$. So the sum is $\frac{2\cos A\mathrm{cosec}A}{{\cot }^{2}A}=2\cdot \frac{\cos A}{\sin A}\cdot \frac{{\sin }^{2}A}{{\cos }^{2}A}=\frac{2\sin A}{\cos A}=2\tan A$. Writing ${\mathrm{cosec}}^{2}A-1={\tan }^{2}A$ instead of ${\cot }^{2}A$ is what produces $2\cot A$. Check at $A=60$ deg: the sum is about 3.464 and $2\tan 60^{\circ }\approx 3.464$.