(a)OR(b)

For acute angles A and B, if sec $\displaystyle (2 \mathrm{~A}-\mathrm{B})=\sqrt{2}$ and $\displaystyle \operatorname{cosec}(\mathrm{A}+\mathrm{B})=2$, then find the values of A and B.

$\displaystyle \sec (2 A-B)=\sqrt{2} \Rightarrow 2 A-B=45^{\circ}$ $\displaystyle \operatorname{cosec}(\mathrm{A}+\mathrm{B})=2 \Rightarrow \mathrm{~A}+\mathrm{B}=30^{\circ}$ On solving, $\displaystyle \mathrm{A}=25^{\circ}, \mathrm{B}=5^{\circ}$ Evaluate: $\displaystyle \frac{2 \cos 30^{\circ}-\cot ^{3} 60^{\circ}}{\tan 30^{\circ}}$ \[\begin{array}{l} \frac{2 \cos 30^{\circ}-\cot ^{3} 60^{\circ}}{\tan 30^{\circ}}=\frac{2 \times \dfrac{\sqrt{3}}{2}-\left(\dfrac{1}{\sqrt{3}}\right)^{3}}{\dfrac{1}{\sqrt{3}}} \\ =\frac{8}{3} \end{array} \]