To determine the expression equivalent to \( \frac{\sin(1)\cos(17) - \cos(1)\sin(77)}{12} \):
1. Use the trigonometric identity: \( \sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b) \).
2. Apply the identity to simplify the given expression.
3. Substitute the values of \(a\) and \(b\) as \(1\) and \(17\) for the first term and as \(1\) and \(77\) for the second term in the expression.
4. Evaluate the trigonometric functions at those angles.
5. Combine the terms and divide the result by \(12\).
By following these steps, you can find the expression equivalent to \( \frac{\sin(1)\cos(17) - \cos(1)\sin(77)}{12} \) and simplify it accordingly.
To simplify the expression \( \frac{\sin(1) \cos(17) - \cos(1) \sin(77)}{12} \):
1. Use the trigonometric identity: \( \sin(a-b) = \sin(a)\cos(b) - \cos(a)\sin(b) \).
2. Apply the identity to rewrite the expression as \( \sin(1-77) \).
3. Calculate \( \sin(-76) \), which is equivalent to \( -\sin(76) \) due to the periodicity of the sine function.
4. Therefore, the simplified expression is \( -\frac{\sin(76)}{12} \).
