(A) Prove that (cos A + sin A -1)/(cos A - sin A +1)= cosec A - cot A

\[\begin{aligned} \mathrm{LHS} & =\frac{\cos \mathrm{A}+\sin \mathrm{A}-1}{\cos \mathrm{~A}-\sin \mathrm{A}+1} \\ & =\frac{\cot \mathrm{A}+1-\operatorname{cosec} \mathrm{A}}{\cot \mathrm{~A}-1+\operatorname{cosec} \mathrm{A}} \\ & =\frac{\cot \mathrm{A}-\operatorname{cosec} \mathrm{A}+\operatorname{cosec}^{2} \mathrm{~A}-\cot ^{2} \mathrm{~A}}{\cot \mathrm{~A}-1+\operatorname{cosec} \mathrm{A}} \\ & =\frac{(\operatorname{cosec} \mathrm{A}-\cot \mathrm{A})(-1+\operatorname{cosec} \mathrm{A}+\cot \mathrm{A})}{\cot \mathrm{A}-1+\operatorname{cosec} \mathrm{A}} \\ & =\operatorname{cosec} \mathrm{A}-\cot \mathrm{A}=\mathrm{RHS} \end{aligned} \]

\[\begin{aligned} \text { LHS } & =p^{2}-q^{2} \\ & =(\cot \theta+\cos \theta)^{2}-(\cot \theta-\cos \theta)^{2} \\ & =[(\cot \theta+\cos \theta)+(\cot \theta-\cos \theta)][(\cot \theta+\cos \theta)-(\cot \theta-\cos \theta)] \\ & =2 \cot \theta \times 2 \cos \theta=4 \cot \theta \cos \theta \end{aligned} \]

\[\begin{aligned} \text { RHS } & =4 \sqrt{p q} \\ & =4 \sqrt{(\cot \theta+\cos \theta)(\cot \theta-\cos \theta)} \\ & =4 \sqrt{\cot ^{2} \theta-\cos ^{2} \theta} \\ & =4 \sqrt{\cos ^{2} \theta\left(\operatorname{cosec}^{2} \theta-1\right)} \\ & =4 \sqrt{\cos ^{2} \theta \times \cot ^{2} \theta} \\ & =4 \cot \theta \cos \theta \end{aligned} \]

\[\therefore \text { LHS }=\text { RHS } \]