Question

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\csc A+\cot A$, using the identity ${\csc }^{2}A=1+{\cot }^{2}A$

Answer 1

Simplifying the LHS of $\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\csc A+\cot A$

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\frac{\sin A\left(\cos A-\sin A+1\right)}{\sin A\left(\cos A+\sin A-1\right)}$

$=\frac{\sin A\cos A-{\sin }^{2}A+\sin A}{\sin A\left(\cos A+\sin A-1\right)}$

$=\frac{\sin A\cos A+\sin A-\left(1-{\cos }^{2}A\right)}{\sin A\left(\cos A+\sin A-1\right)}$

$=\frac{\sin A\left(\cos A+1\right)-\left(1-\cos A\right)\left(1+\cos A\right)}{\sin A\left(\cos A+\sin A-1\right)}$

$=\frac{\left(1+\cos A\right)\left(\sin A+\cos A-1\right)}{\sin A\left(\cos A+\sin A-1\right)}$

$=\frac{1+\cos A}{\sin A}$

$=\frac{1}{\sin A}+\frac{\cos A}{\sin A}$

$=\csc A+\cot A$

$=RHS$