We have $\frac{\cos \left(45^{\circ}\right)}{\sec \left(30^{\circ}\right)+\ cosec \left(30^{\circ}\right)}$ $=\frac{\frac{1}{\sqrt{2}}}{\frac{2}{\sqrt{3}}+2}$ $=\frac{\frac{1}{\sqrt{2}}}{2\left(\frac{1}{\sqrt{3}}+1\right)}$ $=\frac{1}{2 \sqrt{2}\left(\frac{1+\sqrt{3}}{\sqrt{3}}\right)}$ $=\frac{\sqrt{3}}{2 \sqrt{2}(1+\sqrt{3})}$ it is clear that the denominator has an irrational number, we need to rationalize it, we get $=\frac{\sqrt{3}}{2 \sqrt{2}(1+\sqrt{3})} \times \frac{\sqrt{2}(1-\sqrt{3})}{\sqrt{2}(1-\sqrt{3})}$ $=\frac{\sqrt{2}(\sqrt{3}-3)}{2(2)\left(1^{2}-(\sqrt{3})^{2}\right)}$ $=\frac{\sqrt{6}-3 \sqrt{2}}{4(1-3)}$ $=\frac{\sqrt{6}-3 \sqrt{2}}{4(-2)}$ $=\frac{\sqrt{6}-3 \sqrt{2}}{-8}$ $=\frac{3 \sqrt{2}-\sqrt{6}}{8}$
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