Verify each of the following:

(i) sin 60° cos 30° — cos 60° sin 30° = sin 30°

(ii) cos 60° cos 30° + sin 60° sin 30° = cos 30°

(iii) 2 sin 30° cos 30° = sin 60°

(iv) 2 sin 45° cos 45° = sin 90°

(i) Consider L.H.S. = sin 60° cos 30° — cos 60° sin 30°

= (√3/2) × (√3/2) – (1/2)(1/2)

= (3/4) – (1/4)

= 2/4

=1/2

Consider R.H.S. = sin 30° = 1/2

L.H.S. = R.H.S.

Hence, verified.

(ii) Consider L.H.S. = cos 60° cos 30° + sin 60° sin 30°

= (1/2) × (√3/2) + (√3/2)(1/2)

= (√3/4) + (√3/4)

= √3/2 = cos 30° = R.H.S.

∴ L.H.S. = R.H.S.

Hence, verified.

(iii) Consider L.H.S. = 2 sin 30° cos 30°

= 2 × (1/2) × (√3/2)

= √3/2 = sin 60° = R.H.S.

∴ L.H.S. = R.H.S.

Hence, verified.

(iv) Consider L.H.S. = 2 sin 45° cos 45°

= 2 × (1/√2) × (1/√2)

= (2 × 1/2)

= 1 = sin 90° = R.H.S.

∴ L.H.S. = R.H.S.

Hence, verified.