Find `1-sin10^(@)sin50^(@)sin70^(@)`.

Answer

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

### Step 1: Identify the formula
We will use the trigonometric identity:
\[ \sin \theta \cdot \sin (60^\circ - \theta) \cdot \sin (60^\circ + \theta) = \frac{1}{4} \sin 3\theta \]
In this case, we can set \( \theta = 10^\circ \). Thus, we have:
\[ \sin 10^\circ \cdot \sin (60^\circ - 10^\circ) \cdot \sin (60^\circ + 10^\circ) = \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ \]

### Step 2: Substitute the values into the formula
Using the identity:
\[ \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{4} \sin 30^\circ \]

### Step 3: Calculate \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]
So, substituting this value into our equation gives:
\[ \sin 10^\circ \cdot \sin 50^\circ \cdot \sin 70^\circ = \frac{1}{4} \cdot \frac{1}{2} = \frac{1}{8} \]

### Step 4: Calculate \( 1 - \sin 10^\circ \sin 50^\circ \sin 70^\circ \)
Now we can find:
\[ 1 - \sin 10^\circ \sin 50^\circ \sin 70^\circ = 1 - \frac{1}{8} \]

### Step 5: Simplify the expression
To simplify:
\[ 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8} \]

### Final Answer
Thus, the final answer is:
\[ \boxed{\frac{7}{8}} \]

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