We know that,
\(\cos 45^{\circ}=\frac{1}{\sqrt{2}}, \sin 60^{\circ}=\frac{\sqrt{3}}{2}, \sec 30^{\circ}=\frac{2}{\sqrt{3}}, \text{cosec}30^\circ = 2\)
\(\therefore \frac{\cos 45^{\circ}+\sin 60^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}\)
\(=\cfrac{\frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}+2}\)
\(=\cfrac{\frac{2+\sqrt{5}}{2 \sqrt{2}}}{\frac{2+2 \sqrt{3}}{\sqrt{3}}}\)
\(=\frac{2+\sqrt{5} }{2 \sqrt{2}}\times \frac{\sqrt{3}}{2+2 \sqrt{3}}\)
\(= \frac{2 \sqrt{3}+\sqrt{15}}{4\sqrt 2+4 \sqrt{6}}\)
