12sin(5π/12)cos(5π/12)×cos25°/sin(-115°)

To simplify the expression \( 12 \sin\left(\frac{5\pi}{12}\right) \cos\left(\frac{5\pi}{12}\right) \times \frac{\cos(25^\circ)}{\sin(-115^\circ)} \), we can use some trigonometric identities and properties.

Step 1: Simplify \(\sin(-115^\circ)\)

Using the property \(\sin(-\theta) = -\sin(\theta)\):

\[ \sin(-115^\circ) = -\sin(115^\circ) \]

Step 2: Use the double angle formula for sine

We know that:

\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]

In our expression, we can let \(\theta = \frac{5\pi}{12}\):

\[ \sin\left(2 \cdot \frac{5\pi}{12}\right) = \sin\left(\frac{5\pi}{6}\right) \]

Calculate \(\sin\left(\frac{5\pi}{6}\right)\)

The sine of \(\frac{5\pi}{6}\) is:

\[ \sin\left(\frac{5\pi}{6}\right) = \sin(180^\circ - 30^\circ) = \sin(30^\circ) = \frac{1}{2} \]

So:

\[ 12 \sin\left(\frac{5\pi}{12}\right) \cos\left(\frac{5\pi}{12}\right) = 12 \cdot \frac{1}{2} = 6 \]

Step 3: Replace \(\sin(-115^\circ)\)

Now substituting back, we have:

\[ \sin(-115^\circ) = -\sin(115^\circ) \]

Thus:

\[ \frac{\cos(25^\circ)}{\sin(-115^\circ)} = \frac{\cos(25^\circ)}{-\sin(115^\circ)} = -\frac{\cos(25^\circ)}{\sin(115^\circ)} \]

Step 4: Use \(\sin\) identity

We can use the identity \( \sin(115^\circ) = \cos(25^\circ) \):

\[ -\frac{\cos(25^\circ)}{\sin(115^\circ)} = -\frac{\cos(25^\circ)}{\cos(25^\circ)} = -1 \quad \text{(as long as } \cos(25^\circ) \neq 0\text{)} \]

Step 5: Combine everything

Now combining everything together:

\[ 6 \times -1 = -6 \]

Thus, the final result for the expression \( 12 \sin\left(\frac{5\pi}{12}\right) \cos\left(\frac{5\pi}{12}\right) \times \frac{\cos(25^\circ)}{\sin(-115^\circ)} \) is:

\[ \boxed{-6} \]