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

Options

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

Explanation

To solve the problem \( \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \), we can use the sine subtraction formula. Here’s a step-by-step solution:

### Step 1: Apply the sine subtraction formula
We know that:
\[ \sin x - \sin y = 2 \cos\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) \]
Let \( x = 50^\circ \) and \( y = 70^\circ \).

### Step 2: Calculate \( \sin 50^\circ - \sin 70^\circ \)
Using the formula:
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos\left(\frac{50^\circ + 70^\circ}{2}\right) \sin\left(\frac{50^\circ - 70^\circ}{2}\right) \]
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,
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos(60^\circ) \sin(-10^\circ) \]

### 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 expression
This simplifies to:
\[ 0 \]

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