(a) Prove the following trigonometric identity: (1+ cosec A)/(cosec A)=(cos^2 A)/(1- sin A)

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

\[\begin{aligned} & \sin (2 A+B)=\frac{\sqrt{3}}{2} \Rightarrow 2 A+B=60^{\circ} \quad \text { --- (1) } \\ & \tan (A+2 B)=1 \Rightarrow A+2 B=45^{\circ} \quad \text { --- (2) } \\ & \text { Solving (1) \& (2), we get } A=25^{\circ} \text { and } B=10^{\circ} \\ & \begin{aligned} \cot (4 A-7 B) & =\cot 30^{\circ} \\ & =\sqrt{3} \end{aligned} \end{aligned} \]