Find the value of \[ \left(\frac{\sin 35^{\circ}}{\cos 55^{\circ}}\right)^2+\left(\frac{\cos 55^{\circ}}{\sin 35^{\circ}}\right)^2-2 \cos 30^{\circ} . \]

Solution

Given:
\[ \left(\frac{\sin 35^\circ}{\cos 55^\circ}\right)^2 + \left(\frac{\cos 55^\circ}{\sin 35^\circ}\right)^2 – 2\cos 30^\circ \]

Step 1: Use the identity \(\cos(90^\circ – \theta) = \sin \theta\)

\[ \left(\frac{\sin 35^\circ}{\cos 55^\circ}\right)^2 + \left(\frac{\cos 55^\circ}{\sin 35^\circ}\right)^2 – 2\cos 30^\circ \]
\[ = \left(\frac{\sin 35^\circ}{\sin(90^\circ – 35^\circ)}\right)^2 + \left(\frac{\sin(90^\circ – 55^\circ)}{\sin 35^\circ}\right)^2 – 2\cos 30^\circ \]
\[ = \left(\frac{\sin 35^\circ}{\sin 35^\circ}\right)^2 + \left(\frac{\sin 35^\circ}{\sin 35^\circ}\right)^2 – 2\cos 30^\circ \]

Step 2: Simplify

\[ = 1 + 1 – 2\left(\frac{\sqrt{3}}{2}\right) \]
\[ = 2 – \sqrt{3} \]

Conclusion

The value of the given expression is \(2 – \sqrt{3}\).