(i) Given assin 12x + sin 4x
On using the formula,
sin A + sin B = 2 sin (A + B)/2 cos (A - B)/2
sin 12x + sin 4x = 2 sin (12x + 4x)/2 cos (12x – 4x)/2
= 2 sin 16x/2 cos 8x/2
= 2 sin 8x cos 4x
(ii)sin 5x – sin x
On using the formula,
sin A – sin B = 2 cos (A + B)/2 sin (A - B)/2
sin 5x – sin x = 2 cos (5x + x)/2 sin (5x – x)/2
= 2 cos 6x/2 sin 4x/2
= 2 cos 3x sin 2x
(iii)cos 12x + cos 8x
On using the formula,
cos A + cos B = 2 cos (A + B)/2 cos (A - B)/2
cos 12x + cos 8x = 2 cos (12x + 8x)/2 cos (12x – 8x)/2
= 2 cos 20x/2 cos 4x/2
= 2 cos 10x cos 2x
(iv)cos 12x – cos 4x
On using the formula,
cos A – cos B = -2 sin (A + B)/2 sin (A - B)/2
cos 12x – cos 4x = -2 sin (12x + 4x)/2 sin (12x – 4x)/2
= -2 sin 16x/2 sin 8x/2
= -2 sin 8x sin 4x
(v) Given as sin 2x + cos 4x
sin 2x + cos 4x = sin 2x + sin (90° – 4x)
On using the formula,
sin A + sin B = 2 sin (A + B)/2 cos (A - B)/2
sin 2x + sin (90° – 4x) = 2 sin (2x + 90° – 4x)/2 cos (2x – 90° + 4x)/2
= 2 sin (90° – 2x)/2 cos (6x – 90°)/2
= 2 sin (45° – x) cos (3x – 45°)
= 2 sin (45° – x) cos {-(45° – 3x)} (since, {cos (-x) = cos x})
= 2 sin (45° – x) cos (45° – 3x)
= 2 sin (π/4 – x) cos (π/4 – 3x)
