Prove that \(2\sin\frac{5\pi}{12}\cos\frac{\pi}{12}=\frac{\sqrt{3}+1}{2}\)
Solution
Using the identity:
\[
2\sin A\cos B=\sin(A+B)+\sin(A-B)
\]
\[
2\sin\frac{5\pi}{12}\cos\frac{\pi}{12}
\]
\[
= \sin\left(\frac{5\pi}{12}+\frac{\pi}{12}\right)
+\sin\left(\frac{5\pi}{12}-\frac{\pi}{12}\right)
\]
\[
= \sin\frac{6\pi}{12}+\sin\frac{4\pi}{12}
\]
\[
= \sin\frac{\pi}{2}+\sin\frac{\pi}{3}
\]
\[
= 1+\frac{\sqrt{3}}{2}
\]
\[
= \frac{2+\sqrt{3}}{2}
\]
Hence Proved
\[
2\sin\frac{5\pi}{12}\cos\frac{\pi}{12}
=\frac{2+\sqrt{3}}{2}
\]