The value of `sin50^(@)-sin70^(@)+sin10^(@)` is

Options

A. 1
B. 0
C. `(1)/(2)`
D. 2

Explanation

To find the value of \( \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \), we can follow these steps:

### Step 1: Use the sine difference formula
We know that the difference of two sines can be expressed using the formula:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Here, let \( A = 50^\circ \) and \( B = 70^\circ \).

### Step 2: Apply the formula
Substituting the values of \( A \) and \( B \):
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos\left(\frac{50^\circ + 70^\circ}{2}\right) \sin\left(\frac{50^\circ - 70^\circ}{2}\right) \]
Calculating the averages:
\[ \frac{50^\circ + 70^\circ}{2} = \frac{120^\circ}{2} = 60^\circ \]
\[ \frac{50^\circ - 70^\circ}{2} = \frac{-20^\circ}{2} = -10^\circ \]
Thus, we can rewrite the equation:
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos(60^\circ) \sin(-10^\circ) \]

### Step 3: Substitute known values
We know that:
\[ \cos(60^\circ) = \frac{1}{2} \quad \text{and} \quad \sin(-10^\circ) = -\sin(10^\circ) \]
Substituting these values:
\[ \sin 50^\circ - \sin 70^\circ = 2 \cdot \frac{1}{2} \cdot (-\sin(10^\circ)) = -\sin(10^\circ) \]

### Step 4: Combine with \( \sin 10^\circ \)
Now we substitute back into the original expression:
\[ \sin 50^\circ - \sin 70^\circ + \sin 10^\circ = -\sin(10^\circ) + \sin(10^\circ) \]

### Step 5: Simplify
The terms cancel out:
\[ -\sin(10^\circ) + \sin(10^\circ) = 0 \]

### Final Answer
Thus, the value of \( \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \) is:
\[ \boxed{0} \]