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: \[ \boxed{-6} \]