The value of` sin50^9(@) – sin 70^(@) + sin 10^(@)` is equal to

Options

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

Answer

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

### Step 1: Identify the expression
We start with the expression:
\[ \sin 50^\circ - \sin 70^\circ + \sin 10^\circ \]

### Step 2: Apply the sine subtraction identity
We use the identity:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Let \( A = 50^\circ \) and \( B = 70^\circ \). Then we can rewrite \( \sin 50^\circ - \sin 70^\circ \):
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos\left(\frac{50^\circ + 70^\circ}{2}\right) \sin\left(\frac{50^\circ - 70^\circ}{2}\right) \]

### Step 3: Calculate \( \frac{A + B}{2} \) and \( \frac{A - B}{2} \)
Calculating \( \frac{50^\circ + 70^\circ}{2} \):
\[ \frac{50^\circ + 70^\circ}{2} = \frac{120^\circ}{2} = 60^\circ \]
Calculating \( \frac{50^\circ - 70^\circ}{2} \):
\[ \frac{50^\circ - 70^\circ}{2} = \frac{-20^\circ}{2} = -10^\circ \]

### Step 4: Substitute back into the equation
Now substituting these values back into the identity:
\[ \sin 50^\circ - \sin 70^\circ = 2 \cos(60^\circ) \sin(-10^\circ) \]

### Step 5: Simplify using known values
We know:
\[ \cos(60^\circ) = \frac{1}{2} \quad \text{and} \quad \sin(-10^\circ) = -\sin(10^\circ) \]
Thus, we have:
\[ \sin 50^\circ - \sin 70^\circ = 2 \cdot \frac{1}{2} \cdot (-\sin 10^\circ) = -\sin 10^\circ \]

### Step 6: Combine with \( \sin 10^\circ \)
Now substituting back into the original expression:
\[ \sin 50^\circ - \sin 70^\circ + \sin 10^\circ = -\sin 10^\circ + \sin 10^\circ = 0 \]

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