express the sum or difference as a product. if possible, find the exact value of this product. \n$$ sin…

express the sum or difference as a product. if possible, find the exact value of this product. \n$$ sin left( \frac { 7 pi } { 12 } \right) - sin left( \frac { pi } { 12 } \right) $$ \nexpress the difference as a product. \n$$ sin left( \frac { 7 pi } { 12 } \right) - sin left( \frac { pi } { 12 } \right) = - \frac { sqrt { 2 } } { 2 } $$

express the sum or difference as a product. if possible, find the exact value of this product. \n$$ sin left( \frac { 7 pi } { 12 } \right) - sin left( \frac { pi } { 12 } \right) $$ \nexpress the difference as a product. \n$$ sin left( \frac { 7 pi } { 12 } \right) - sin left( \frac { pi } { 12 } \right) = - \frac { sqrt { 2 } } { 2 } $$

Answer

Explanation:

Step1: Apply the sine subtraction formula

The formula for (\sin A-\sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}). Here (A=\frac{7\pi}{12}) and (B=\frac{\pi}{12}). [ \begin{align*} \sin\frac{7\pi}{12}-\sin\frac{\pi}{12}&=2\cos\frac{\frac{7\pi}{12}+\frac{\pi}{12}}{2}\sin\frac{\frac{7\pi}{12}-\frac{\pi}{12}}{2}\ &=2\cos\frac{\frac{8\pi}{12}}{2}\sin\frac{\frac{6\pi}{12}}{2}\ &=2\cos\frac{2\pi}{6}\sin\frac{\pi}{4} \end{align*} ]

Step2: Simplify the trigonometric values

We know that (\cos\frac{\pi}{3}=\frac{1}{2}) and (\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}). [ \begin{align*} 2\cos\frac{\pi}{3}\sin\frac{\pi}{4}&=2\times\frac{1}{2}\times\frac{\sqrt{2}}{2}\ &=\frac{\sqrt{2}}{2} \end{align*} ]

Answer:

(\sin\frac{7\pi}{12}-\sin\frac{\pi}{12}=\frac{\sqrt{2}}{2})