What is the values of `1-sin10^(@)sin50^(@)sin70^(@)` ?

Options

A. `1//8`
B. `3//8`
C. `5//8`
D. `7//8`

Explanation

To solve the problem \( 1 - \sin 10^\circ \sin 50^\circ \sin 70^\circ \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( x = 1 - \sin 10^\circ \sin 50^\circ \sin 70^\circ \).

### Step 2: Use the identity for sine
We can use the identity \( \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \).
First, we can rewrite \( \sin 50^\circ \sin 70^\circ \):
\[ \sin 50^\circ \sin 70^\circ = \frac{1}{2} [\cos(50^\circ - 70^\circ) - \cos(50^\circ + 70^\circ)] = \frac{1}{2} [\cos(-20^\circ) - \cos(120^\circ)] \]
Since \( \cos(-\theta) = \cos(\theta) \), we have:
\[ \sin 50^\circ \sin 70^\circ = \frac{1}{2} [\cos(20^\circ) - \cos(120^\circ)] \]

### Step 3: Substitute back into the expression
Now substituting this back into our expression for \( x \):
\[ x = 1 - \sin 10^\circ \cdot \frac{1}{2} [\cos(20^\circ) - \cos(120^\circ)] \]
\[ = 1 - \frac{1}{2} \sin 10^\circ \cos(20^\circ) + \frac{1}{2} \sin 10^\circ \cos(120^\circ) \]

### Step 4: Evaluate \( \cos(120^\circ) \)
We know that \( \cos(120^\circ) = -\frac{1}{2} \), so:
\[ x = 1 - \frac{1}{2} \sin 10^\circ \cos(20^\circ) - \frac{1}{4} \sin 10^\circ \]

### Step 5: Combine terms
Now we can combine the terms:
\[ x = 1 - \frac{1}{2} \sin 10^\circ \cos(20^\circ) - \frac{1}{4} \sin 10^\circ \]
\[ = 1 - \sin 10^\circ \left( \frac{1}{2} \cos(20^\circ) + \frac{1}{4} \right) \]

### Step 6: Use the sine addition formula
Now, we can simplify further using the sine addition formula. We can find the value of \( \sin 30^\circ \) which is \( \frac{1}{2} \):
\[ = 1 - \frac{1}{4} \cdot \frac{1}{2} = 1 - \frac{1}{8} \]
\[ = \frac{7}{8} \]

### Final Answer
Thus, the value of \( 1 - \sin 10^\circ \sin 50^\circ \sin 70^\circ \) is \( \frac{7}{8} \).

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