which expression is equivalent to $sinleft(\f...
which expression is equivalent to $sinleft(\frac{pi}{12}\right)cosleft(\frac{7pi}{12}\right)-cosleft(\frac{pi}{12}\right)sinleft(\frac{7pi}{12}\right)$?\n$cosleft(-\frac{pi}{2}\right)$\n$sinleft(-\frac{pi}{2}\right)$\n$cosleft(\frac{2pi}{3}\right)$\n$sinleft(\frac{2pi}{3}\right)$
Answer
# Answer:
B. $\sin\left(-\frac{\pi}{2}\right)$
# Explanation:
## Step1: Recall the sine - difference formula
The formula for $\sin(A - B)=\sin A\cos B-\cos A\sin B$.
In the given expression $\sin\left(\frac{\pi}{12}\right)\cos\left(\frac{7\pi}{12}\right)-\cos\left(\frac{\pi}{12}\right)\sin\left(\frac{7\pi}{12}\right)$, we have $A = \frac{\pi}{12}$ and $B=\frac{7\pi}{12}$.
## Step2: Calculate $A - B$
$A - B=\frac{\pi}{12}-\frac{7\pi}{12}=\frac{\pi - 7\pi}{12}=\frac{- 6\pi}{12}=-\frac{\pi}{2}$.
So, $\sin\left(\frac{\pi}{12}\right)\cos\left(\frac{7\pi}{12}\right)-\cos\left(\frac{\pi}{12}\right)\sin\left(\frac{7\pi}{12}\right)=\sin\left(-\frac{\pi}{2}\right)$.