Show that $(1+\cot A-\mathrm{cosec}A)(1+\tan A+\sec A)$ is a constant.
What is that constant?

Write both brackets over sin and cos: $\frac{\sin A+\cos A-1}{\sin A}\cdot \frac{\cos A+\sin A+1}{\cos A}=\frac{(\sin A+\cos A{)}^2-1}{\sin A\cos A}=\frac{1+2\sin A\cos A-1}{\sin A\cos A}=2$. Answering 1 usually means $(\sin A+\cos A{)}^2$ was taken as ${\sin }^{2}A+{\cos }^{2}A$, dropping the cross term. Check at $A=45$ deg: $(1+1-\sqrt{2})(1+1+\sqrt{2})=4-2=2$.