Question

CMR \(cos\dfrac{\pi}{5}-cos\dfrac{2\pi}{5}=\dfrac{1}{2}\)

Answer 1

\(cos\dfrac{\pi}{5}-cos\dfrac{2\pi}{5}\)

\(=-2.sin\dfrac{3\pi}{10}.sin\left(-\dfrac{\pi}{10}\right)\)

\(=2.sin\left(\dfrac{1}{2}-\dfrac{\pi}{5}\right).sin\dfrac{\pi}{10}\)

\(=2.sin\dfrac{\pi}{10}.cos\dfrac{\pi}{5}=\dfrac{sin\dfrac{\pi}{5}.cos\dfrac{\pi}{5}}{cos\dfrac{\pi}{10}}\)

\(=\dfrac{\dfrac{1}{2}sin\dfrac{2\pi}{5}}{cos\left(\dfrac{\pi}{2}-\dfrac{2\pi}{5}\right)}=\dfrac{\dfrac{1}{2}.sin\dfrac{2\pi}{5}}{sin\dfrac{2\pi}{5}}\)\(=\dfrac{1}{2}\)

Answer 2

=−2.sin3π10.sin(−π10)=−2.sin3π10.sin(−π10)

=2.sinπ10.cosπ5=sinπ5.cosπ5cosπ10=2.sinπ10.cosπ5=sinπ5.cosπ5cosπ10

=12