Question : The value of $\frac{\sin ^2 30^{\circ}+\cos ^2 60^{\circ}-\sec 35^{\circ} \cdot \sin 55^{\circ}}{\sec 60^{\circ}+\operatorname{cosec} 30^{\circ}}$ is equal to:
Option 1: $\frac{1}{8}$
Option 2: $-\frac{1}{4}$
Option 3: $\frac{1}{4}$
Option 4: $-\frac{1}{8}$
Correct Answer: $-\frac{1}{8}$
Solution :
Given: $\frac{\sin ^2 30^{\circ}+\cos ^2 60^{\circ}-\sec 35^{\circ} \cdot \sin 55^{\circ}}{\sec 60^{\circ}+\operatorname{cosec} 30^{\circ}}$
= $\frac{(\frac{1}{2})^2+(\frac{1}{2})^2 - \sec 35^{\circ} \cdot \sin (90-35)^{\circ}}{2+2}$
= $\frac{(\frac{1}{4})+(\frac{1}{4}) - \sec 35^{\circ} \cdot \cos 35^{\circ}}{2+2}$
= $\frac{(\frac{2}{4}) - (\frac{1}{ \cos 35^{\circ}}) \cdot \cos 35^{\circ}}{4}$
= $\frac{(\frac{2}{4}) - 1}{4}$
= $-\frac{1}{8}$
Hence, the correct answer is $-\frac{1}{8}$.
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