What is the value of `(sin 45^@)/(sec30^@ + cosec 30^@) ? `

Options

A. `(sqrt3 - 1)`
B. `(sqrt3 (sqrt3 - 1))/(4sqrt2)`
C. `4sqrt2`
D. `sqrt3 (sqrt3 - 1)`

Answer

To find the value of \((\sin 45^\circ) / (\sec 30^\circ + \csc 30^\circ)\), we will follow these steps:

### Step 1: Calculate \(\sin 45^\circ\)
The value of \(\sin 45^\circ\) is known to be:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \]

### Step 2: Calculate \(\sec 30^\circ\)
The secant function is the reciprocal of the cosine function. Therefore:
\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} \]
The value of \(\cos 30^\circ\) is \(\frac{\sqrt{3}}{2}\), so:
\[ \sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 3: Calculate \(\csc 30^\circ\)
The cosecant function is the reciprocal of the sine function. Hence:
\[ \csc 30^\circ = \frac{1}{\sin 30^\circ} \]
The value of \(\sin 30^\circ\) is \(\frac{1}{2}\), so:
\[ \csc 30^\circ = \frac{1}{\frac{1}{2}} = 2 \]

### Step 4: Combine \(\sec 30^\circ\) and \(\csc 30^\circ\)
Now we can add \(\sec 30^\circ\) and \(\csc 30^\circ\):
\[ \sec 30^\circ + \csc 30^\circ = \frac{2}{\sqrt{3}} + 2 \]
To add these fractions, we need a common denominator. The common denominator is \(\sqrt{3}\):
\[ \sec 30^\circ + \csc 30^\circ = \frac{2}{\sqrt{3}} + \frac{2\sqrt{3}}{\sqrt{3}} = \frac{2 + 2\sqrt{3}}{\sqrt{3}} = \frac{2(1 + \sqrt{3})}{\sqrt{3}} \]

### Step 5: Substitute back into the original expression
Now we substitute back into the original expression:
\[ \frac{\sin 45^\circ}{\sec 30^\circ + \csc 30^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{2(1 + \sqrt{3})}{\sqrt{3}}} \]
This can be simplified by multiplying by the reciprocal:
\[ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2(1 + \sqrt{3})} = \frac{\sqrt{3}}{2\sqrt{2}(1 + \sqrt{3})} \]

### Step 6: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by \(1 - \sqrt{3}\):
\[ = \frac{\sqrt{3}(1 - \sqrt{3})}{2\sqrt{2}((1 + \sqrt{3})(1 - \sqrt{3}))} \]
Calculating the denominator:
\[ (1 + \sqrt{3})(1 - \sqrt{3}) = 1 - 3 = -2 \]
Thus, we have:
\[ = \frac{\sqrt{3}(1 - \sqrt{3})}{-4\sqrt{2}} = -\frac{\sqrt{3}(1 - \sqrt{3})}{4\sqrt{2}} \]

### Final Answer
The value of \(\frac{\sin 45^\circ}{\sec 30^\circ + \csc 30^\circ}\) simplifies to:
\[ -\frac{\sqrt{3}(1 - \sqrt{3})}{4\sqrt{2}} \]