sin7π/12 -sinπ/12

To find \( \sin\left(\frac{7\pi}{12}\right) - \sin\left(\frac{\pi}{12}\right) \), we can use the sine angle subtraction formula:

\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]

Let \( A = \frac{7\pi}{12} \) and \( B = \frac{\pi}{12} \). Now we calculate \( A + B \) and \( A - B \):

\[ A + B = \frac{7\pi}{12} + \frac{\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3} \]

\[ A - B = \frac{7\pi}{12} - \frac{\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \]

Now, we can substitute these values into the sine subtraction formula:

\[ \sin\left(\frac{7\pi}{12}\right) - \sin\left(\frac{\pi}{12}\right) = 2 \cos\left(\frac{2\pi}{3} \cdot \frac{1}{2}\right) \sin\left(\frac{\pi}{2} \cdot \frac{1}{2}\right) \]

Calculating the cosine and sine values:

\[ \cos\left(\frac{2\pi}{3} \cdot \frac{1}{2}\right) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

\[ \sin\left(\frac{\pi}{2} \cdot \frac{1}{2}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]

Putting it all together:

\[ \sin\left(\frac{7\pi}{12}\right) - \sin\left(\frac{\pi}{12}\right) = 2 \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{2} \]

Thus, the final result is:

\[ \sin\left(\frac{7\pi}{12}\right) - \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{2}}{2} \]