**(a)** $\sec A=\sqrt2 \Rightarrow A=45° \Rightarrow \sin A=\frac{1}{\sqrt2}$. $\tan B=\sqrt3 \Rightarrow B=60° \Rightarrow \cos B=\frac12$. $$2\sin A\cos B=2\times\frac{1}{\sqrt2}\times\frac12=\frac{1}{\sqrt2}=\frac{\sqrt2}{2}$$

**(b)** $\cos60°=\frac12 \Rightarrow \cos^360°=\frac18 \Rightarrow 4\times\frac18=\frac12$. $\csc30°=2$. Numerator$=\frac12+2=\frac52$. $\tan30°=\frac1{\sqrt3} \Rightarrow \tan^230°=\frac13$. $$\text{Result}=\frac{5/2}{1/3}=\frac{15}{2}$$