Express each of the following as the sum or difference of sines and cosines:
(i) 2 sin 3x cos x
(ii) 2 cos 3x sin 2x
(iii) 2 sin 4x sin 3x
(iv) 2 cos 7x cos 3x
(i) We know, 2 sin A cos B = sin(A + B) + sin(A – B)
2 sin 3x cos x = sin (3x + x) + sin (3x – x)
= sin (4x) + sin (2x)
= sin 4x + sin 2x
(ii) We know, 2 cos A sin B = sin (A + B) – sin (A – B)
2 cos 3x sin 2x = sin (3x + 2x) – sin (3x – 2x)
= sin (5x) – sin (x)
= sin 5x – sin x
(iii) We know, 2 sin A sin B = cos (A – B) – cos (A + B)
2 sin 4x sin 3x = cos (4x – 3x) – cos (4x + 3x)
= cos (x) – cos (7x)
= cos x – cos 7x
(iv) We know, 2 cos A cos B = cos (A + B) + cos (A – B)
2 sin 3x cos x = cos (7x + 3x) + cos (7x – 3x)
= cos (10x) + cos (4x)
= cos 10x + cos 4x