Evaluate `(sin 30)/(cos 45) xx (sin 45)/(cos 30)`

Options

A. `(sqrt2)/(sqrt3)`
B. `(2)/(sqrt3)`
C. `(1)/(sqrt3)`
D. `(1)/(sqrt2)`

Answer

To evaluate the expression \((\sin 30^\circ / \cos 45^\circ) \times (\sin 45^\circ / \cos 30^\circ)\), we can follow these steps:

### Step 1: Identify the values of trigonometric functions
- \(\sin 30^\circ = \frac{1}{2}\)
- \(\cos 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\sin 45^\circ = \frac{1}{\sqrt{2}}\)
- \(\cos 30^\circ = \frac{\sqrt{3}}{2}\)

### Step 2: Substitute the values into the expression
Now substitute these values into the expression:
\[ \frac{\sin 30^\circ}{\cos 45^\circ} \times \frac{\sin 45^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}} \times \frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}} \]

### Step 3: Simplify each fraction
First, simplify \(\frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}}\):
\[ \frac{1}{2} \div \frac{1}{\sqrt{2}} = \frac{1}{2} \times \frac{\sqrt{2}}{1} = \frac{\sqrt{2}}{2} \]

Next, simplify \(\frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{2}}\):
\[ \frac{1}{\sqrt{2}} \div \frac{\sqrt{3}}{2} = \frac{1}{\sqrt{2}} \times \frac{2}{\sqrt{3}} = \frac{2}{\sqrt{2} \cdot \sqrt{3}} = \frac{2}{\sqrt{6}} \]

### Step 4: Combine the results
Now combine the two simplified results:
\[ \frac{\sqrt{2}}{2} \times \frac{2}{\sqrt{6}} = \frac{\sqrt{2} \cdot 2}{2 \cdot \sqrt{6}} = \frac{\sqrt{2}}{\sqrt{6}} \]

### Step 5: Simplify the final result
To simplify \(\frac{\sqrt{2}}{\sqrt{6}}\):
\[ \frac{\sqrt{2}}{\sqrt{6}} = \frac{\sqrt{2}}{\sqrt{2 \cdot 3}} = \frac{\sqrt{2}}{\sqrt{2} \cdot \sqrt{3}} = \frac{1}{\sqrt{3}} \]

Thus, the final answer is:
\[ \frac{1}{\sqrt{3}} \]