The value of `(sin 38^(@) - cos 68^(@))/( cos 68^(@) + sin 38^(@)` is

Options

A. `sqrt(3) tan 40^(@)`
B. `sqrt(3) tan 8^(@)`
C. `sqrt(3) tan 12^(@)`
D. None of these

Answer

To solve the expression \((\sin 38^\circ - \cos 68^\circ) / (\cos 68^\circ + \sin 38^\circ)\), we can follow these steps:

### Step 1: Use the co-function identity
Recall that \(\cos(90^\circ - \theta) = \sin(\theta)\). Therefore, we can rewrite \(\cos 68^\circ\) as:
\[ \cos 68^\circ = \sin(90^\circ - 68^\circ) = \sin 22^\circ \]
Now, substitute this into the expression:
\[ \frac{\sin 38^\circ - \sin 22^\circ}{\sin 22^\circ + \sin 38^\circ} \]

### Step 2: Apply the sine subtraction and addition formulas
We can use the sine subtraction and addition formulas:
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
and
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \(A = 38^\circ\) and \(B = 22^\circ\).

### Step 3: Calculate \(A + B\) and \(A - B\)
\[ A + B = 38^\circ + 22^\circ = 60^\circ \]
\[ A - B = 38^\circ - 22^\circ = 16^\circ \]

### Step 4: Substitute into the formulas
Now substituting into the formulas:
\[ \sin 38^\circ - \sin 22^\circ = 2 \cos\left(\frac{60^\circ}{2}\right) \sin\left(\frac{16^\circ}{2}\right) = 2 \cos 30^\circ \sin 8^\circ \]
\[ \sin 38^\circ + \sin 22^\circ = 2 \sin\left(\frac{60^\circ}{2}\right) \cos\left(\frac{16^\circ}{2}\right) = 2 \sin 30^\circ \cos 8^\circ \]

### Step 5: Substitute back into the expression
Now substituting these back into our expression:
\[ \frac{2 \cos 30^\circ \sin 8^\circ}{2 \sin 30^\circ \cos 8^\circ} \]
The 2's cancel out:
\[ \frac{\cos 30^\circ \sin 8^\circ}{\sin 30^\circ \cos 8^\circ} \]

### Step 6: Substitute values for sine and cosine
We know:
\[ \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2} \]
Substituting these values:
\[ \frac{\frac{\sqrt{3}}{2} \sin 8^\circ}{\frac{1}{2} \cos 8^\circ} = \frac{\sqrt{3} \sin 8^\circ}{\cos 8^\circ} \]

### Step 7: Simplify to get the final answer
This simplifies to:
\[ \sqrt{3} \tan 8^\circ \]

### Final Answer
Thus, the value of the expression is:
\[ \sqrt{3} \tan 8^\circ \]