(i) \(sin\frac{A}{8}sin\frac{3A}{8}\)
[∵ 2 sin A sin B = cos(A – B) – cos(A + B)
[∵ cos(-θ) = cos θ]
(ii) cos(60° + A) sin(120° + A) = \(\frac{1}{2}\)[2 cos(60° + A) sin(120° + A)] [Multiply and divide by 2]
= \(\frac{1}{2}\)[sin((60° + A) + (120° + A))] – sin((60° + A) – (120° + A))]
[∵ 2 cos A sin B = sin(A + B) – sin(A – B)]
= \(\frac{1}{2}\)[sin(180° + 2A) – sin(60° + A – 120° – A)]
= \(\frac{1}{2}\)[(-sin 2A) – sin(-60°)]
= \(\frac{1}{2}\)[-sin 2A + sin 60°]
= \(\frac{1}{2}\)[-sin 2A + \(\frac{\sqrt{3}}{2}\)]
(iii) \(cos\frac{7A}{3}sin\frac{5A}{3}\)
(iv) cos 7θ sin 3θ = \(\frac{1}{2}\)[sin(7θ + 3θ) – sin(7θ – 3θ)]
= \(\frac{1}{2}\)(sin 10θ – sin 4θ)
