Question : Simplify $\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$
Option 1: $\frac{3 \sqrt{2}+\sqrt{6}}{8}$
Option 2: $\frac{\sqrt{3}}{2 \sqrt{2}-2 \sqrt{6}}$
Option 3: $\frac{3 \sqrt{2}-\sqrt{6}}{8}$
Option 4: $\frac{\sqrt{3}}{2 \sqrt{6}-2 \sqrt{2}}$
Correct Answer: $\frac{3 \sqrt{2}-\sqrt{6}}{8}$
Solution :
$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$
= $\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}$
= $\frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}$
= $\frac{\sqrt{3}}{2\sqrt{2}(\sqrt{3}+1)}\times\frac{\sqrt{2}(\sqrt{3}-1)}{\sqrt{2}(\sqrt{3}-1)}$
= $\frac{3 \sqrt{2}-\sqrt{6}}{8}$
Hence, the correct answer is $\frac{3 \sqrt{2}-\sqrt{6}}{8}$.
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