What is the value of `((cos10^(@)+sin20^(@)))/((cos20^(@)-sin10^(@)))`?

Options

A. `(1)/sqrt(3)`
B. `-(1)/(sqrt(3))`
C. `sqrt(3)`
D. `-sqrt(3)`

Explanation

To solve the expression \(\frac{\cos 10^\circ + \sin 20^\circ}{\cos 20^\circ - \sin 10^\circ}\), we will follow these steps:

### Step 1: Rewrite the Trigonometric Functions
We can use the identity \(\cos(90^\circ - \theta) = \sin(\theta)\) to rewrite some of the terms:
- \(\cos 10^\circ = \sin(90^\circ - 10^\circ) = \sin 80^\circ\)
- \(\cos 20^\circ = \sin(90^\circ - 20^\circ) = \sin 70^\circ\)

Thus, we can rewrite the expression as:
\[ \frac{\sin 80^\circ + \sin 20^\circ}{\sin 70^\circ - \sin 10^\circ} \]

### Step 2: Use the Sum and Difference Formulas
Next, we can apply the sum-to-product identities:
- For the numerator: \(\sin a + \sin b = 2 \sin\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right)\)
- For the denominator: \(\sin a - \sin b = 2 \cos\left(\frac{a+b}{2}\right) \sin\left(\frac{a-b}{2}\right)\)

Applying these identities:
- Numerator:
\[ \sin 80^\circ + \sin 20^\circ = 2 \sin\left(\frac{80^\circ + 20^\circ}{2}\right) \cos\left(\frac{80^\circ - 20^\circ}{2}\right) = 2 \sin(50^\circ) \cos(30^\circ) \]
- Denominator:
\[ \sin 70^\circ - \sin 10^\circ = 2 \cos\left(\frac{70^\circ + 10^\circ}{2}\right) \sin\left(\frac{70^\circ - 10^\circ}{2}\right) = 2 \cos(40^\circ) \sin(30^\circ) \]

### Step 3: Substitute Back into the Expression
Now substituting back into our expression:
\[ \frac{2 \sin(50^\circ) \cos(30^\circ)}{2 \cos(40^\circ) \sin(30^\circ)} \]

### Step 4: Simplify the Expression
The \(2\) cancels out:
\[ \frac{\sin(50^\circ) \cos(30^\circ)}{\cos(40^\circ) \sin(30^\circ)} \]

### Step 5: Substitute Known Values
We know:
- \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
- \(\sin(30^\circ) = \frac{1}{2}\)

Substituting these values:
\[ \frac{\sin(50^\circ) \cdot \frac{\sqrt{3}}{2}}{\cos(40^\circ) \cdot \frac{1}{2}} = \frac{\sin(50^\circ) \cdot \sqrt{3}}{\cos(40^\circ)} \]

### Step 6: Use the Co-Function Identity
Since \(\sin(50^\circ) = \cos(40^\circ)\), we can simplify further:
\[ \frac{\cos(40^\circ) \cdot \sqrt{3}}{\cos(40^\circ)} = \sqrt{3} \]

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