Prove that \(2\cos\frac{5\pi}{12}\cos\frac{\pi}{12}=\frac{1}{2}\)
Solution
Using the identity:
\[
2\cos A\cos B=\cos(A+B)+\cos(A-B)
\]
\[
2\cos\frac{5\pi}{12}\cos\frac{\pi}{12}
\]
\[
= \cos\left(\frac{5\pi}{12}+\frac{\pi}{12}\right)
+\cos\left(\frac{5\pi}{12}-\frac{\pi}{12}\right)
\]
\[
= \cos\frac{6\pi}{12}+\cos\frac{4\pi}{12}
\]
\[
= \cos\frac{\pi}{2}+\cos\frac{\pi}{3}
\]
\[
= 0+\frac{1}{2}
\]
\[
= \frac{1}{2}
\]
Hence Proved
\[
2\cos\frac{5\pi}{12}\cos\frac{\pi}{12}=\frac{1}{2}
\]