(a) 2 sin \(\frac {2\pi}{9}\) cos \(\frac {\pi}{9}\)
= sin (\(\frac {2\pi}{9}\) + \(\frac {\pi}{9}\)) + sin (\(\frac {2\pi}{9}\) - \(\frac {\pi}{9}\))
= sin (\(\frac {3\pi}{9}\)) + sin (\(\frac {\pi}{9}\))
= sin (\(\frac {\pi}{9}\)) + sin (\(\frac {\pi}{9}\))
(b) cos 55° cos 45°
= \(\frac 12 \) (cos (55° + 45°) + cos (55° - 45°))
= \(\frac 12 \) (cos 100° + cos 10°)
(c) sin 6 θ cos 4 θ
= \(\frac 12 \) (sin 6 θ + 4 θ) + sin ( 6 θ - 4 θ)
= \(\frac 12 \) (sin 10 θ + 2 θ)
(d) sin \(\frac {\pi}{9}\) + sin \(\frac {2\pi}{9}\)
= 2 sin \((\frac {\frac {\pi}{9} + \frac {2\pi}{9}}{2}) cos (\frac {\frac \pi9 -\frac {2\pi}{9}}{2})\)
= = 2 sin \((\frac {3\pi}{18}) cos ( \frac {-\pi}{18})\)
(e) sin 55° + sin 45°
= 2 sin \((\frac {55°+45°}{2}) cos (\frac {55°-45°}{2})\)
= 2 sin \((\frac {100°}{2}) cos (\frac {10°}{2})\)
= 2 sin 50° cos 5°
(f) cos \(\frac {3\pi}{2} cos \frac {\pi}{2} \) = 0 x 0
= 0 (∵ \(cos \frac {\pi}{2} = \) 0 & cos \(\frac {3\pi}{2}\) = 0)
(g) cos \(\frac {3\pi}{2} + cos \frac {\pi}{2} \)
= 0 + 0 = 0
or
(f) cos \(\frac {3\pi}{2} cos \frac {\pi}{2} \)
= \(\frac 12 \) ( cos ( \(\frac {3\pi}{2} cos \frac {\pi}{2} \)) + cos ( \(\frac {3\pi}{2} - \frac {\pi}{2} \) ))
= \(\frac 12 \) ( cos ( \((cos 2\pi + cos \pi) = \) \(\frac 12 \) (1 -1) = 0
(g) cos \(\frac {3\pi}{2} + cos \frac {\pi}{2} \)
= 2 cos \(\frac {\frac {3\pi}{2}+\frac {\pi}{2}}{2} cos \frac {\frac {3\pi}{2}-\frac {\pi}{2}}{2} \)
= 2 cos π cos π/2
= 2 x -1 x 0 = 0