Question

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=cosecA+\cot A$, using the identity $cose{c}^{2}A=1.$

Answer 1

Given:

$\Rightarrow$$\frac{1+\cos A-\sin A}{\sin A+\cos A-1}$

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

$=\frac{2{\cos }^2\frac{A}{2}-2\sin \frac{A}{2}\cos \frac{A}{2}}{2\sin \frac{A}{2}\cos \frac{A}{2}-2{\sin }^2\frac{A}{2}}$ Using, $[\because {\cos }^{2}x=\frac{1+\cos 2x}{2}$,$\sin 2x=2\sin x\cos x]$

$=\frac{2\cos \frac{A}{2}(\cos \frac{A}{2}-\sin \frac{A}{2})}{2\sin \frac{A}{2}(\cos \frac{A}{2}-\sin \frac{A}{2})}$

$=\frac{2\cos \frac{A}{2}}{2\sin \frac{A}{2}}$

$=\frac{2{\cos }^2\frac{A}{2}}{2\sin \frac{A}{2}\cos \frac{A}{2}}$

$=\frac{1+\cos A}{\sin A}$ ................. Using, $[\because {\cos }^{2}x=\frac{1+\cos 2x}{2}$,$\sin 2x=2\sin x\cos x]$

$=cosecA+\cot A$