Value of $\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$

Question

I just put the values from the trignometric table to solve, but the answer is different in the answer book.

$$\frac{\cos 45}{\sec 30 + \operatorname{cosec} 30}$$

Answer 1

$$\dfrac{\cos 45}{\sec 30 + \csc 30} = \frac{\cos 45}{\frac 1{\cos 30} + \frac 1{\sin 30}} = \frac{\frac {\sqrt 2}2}{\frac{2}{\sqrt 3} + 2} $$

$$= \frac{\frac {\sqrt 2}2}{\frac{2}{\sqrt 3} + 2} \cdot \frac{2\sqrt 3}{2\sqrt 3} = \frac{\sqrt 2\sqrt 3}{2\left(2+2\sqrt 3\right)}$$

$$=\frac{\sqrt 6}{4(1+\sqrt 3)}\cdot\frac{1 - \sqrt 3}{1-\sqrt 3} = \frac{\sqrt 6(1-\sqrt 3)}{4(1 - 3)} = \frac{\sqrt 6-3\sqrt 2}{-8}$$

$$ = \frac{3\sqrt 2-\sqrt 6}{8}$$