Answer :
To simplify the expression \(\cos 45^\circ + \sin 60^\circ / \sec 30^\circ + \csc 30^\circ\), let's evaluate each trigonometric function involved:
\[
\cos 45^\circ = \frac{1}{\sqrt{2}}
\]
\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\]
\[
\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}
\]
\[
\csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2
\]
Now substitute these values into the expression:
\[
\cos 45^\circ + \frac{\sin 60^\circ}{\sec 30^\circ} + \csc 30^\circ = \frac{1}{\sqrt{2}} + \frac{\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}} + 2
\]
Simplify the fraction:
\[
\frac{\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}} = \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4}
\]
So the expression becomes:
\[
\frac{1}{\sqrt{2}} + \frac{3}{4} + 2
\]
Combine the terms:
\[
\frac{1}{\sqrt{2}} \approx 0.707
\]
\[
\frac{1}{\sqrt{2}} + \frac{3}{4} + 2 \approx 0.707 + 0.75 + 2 = 3.457
\]
Thus, the simplified expression is approximately:
\[
3.457
\]
\[
\cos 45^\circ = \frac{1}{\sqrt{2}}
\]
\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\]
\[
\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}
\]
\[
\csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2
\]
Now substitute these values into the expression:
\[
\cos 45^\circ + \frac{\sin 60^\circ}{\sec 30^\circ} + \csc 30^\circ = \frac{1}{\sqrt{2}} + \frac{\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}} + 2
\]
Simplify the fraction:
\[
\frac{\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}} = \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4}
\]
So the expression becomes:
\[
\frac{1}{\sqrt{2}} + \frac{3}{4} + 2
\]
Combine the terms:
\[
\frac{1}{\sqrt{2}} \approx 0.707
\]
\[
\frac{1}{\sqrt{2}} + \frac{3}{4} + 2 \approx 0.707 + 0.75 + 2 = 3.457
\]
Thus, the simplified expression is approximately:
\[
3.457
\]