Question : The value of $\frac{\sin ^2 30^{\circ}+\cos ^2 60^{\circ}+\sec 45^{\circ} × \sin 45^{\circ}}{\sec 60^{\circ}+{\text{cosec}} 30^{\circ}}$ is:
Option 1: $\frac{1}{4}$
Option 2: $-\frac{3}{8}$
Option 3: $\frac{3}{8}$
Option 4: $-\frac{1}{4}$
Correct Answer: $\frac{3}{8}$
Solution : $\frac{\sin ^2 30^{\circ}+\cos ^2 60^{\circ}+\sec 45^{\circ} × \sin 45^{\circ}}{\sec 60^{\circ}+{\text{cosec}} 30^{\circ}}$ $=\frac{\frac{1}{2}^{2} + \frac{1}{2}^{2} + \sqrt 2 × \frac{1}{\sqrt 2}}{2 + 2}$ $=\frac{\frac{1}{4} + \frac{1}{4} + 1}{2 + 2}$ $=\frac{\frac{3}{2}}{4}$ $=\frac{3}{8}$ Hence, the correct answer is $\frac{3}{8}$.
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