Find the values of :

(i) 7 sin 30° cos 60°

(ii) 3 sin² 45° + 2 cos² 60°

(iii) cos² 45° + sin² 60° + sin² 30°

(iv) cos 90° + cos² 45° sin 30° tan 45°.

(i) 7 sin 30° cos 60°

$= 7 \times \dfrac{1}{2} \times \dfrac{1}{2} \\[1em] = \dfrac{7}{4}.$

Hence, 7 sin 30° cos 60° = $\dfrac{7}{4}$.

(ii) 3 sin² 45° + 2 cos² 60°

$= 3 \times \Big(\dfrac{1}{\sqrt{2}}\Big)^2 + 2 \times \Big(\dfrac{1}{2}\Big)^2 \\[1em] = 3 \times \dfrac{1}{2} + 2 \times \dfrac{1}{4} \\[1em] = \dfrac{3}{2} + \dfrac{1}{2} \\[1em] = \dfrac{4}{2} \\[1em] = 2.$

Hence, 3 sin² 45° + 2 cos² 60° = 2.

(iii) cos² 45° + sin² 60° + sin² 30°

$= \Big(\dfrac{1}{\sqrt{2}}\Big)^2 + \Big(\dfrac{\sqrt{3}}{2}\Big)^2 + \Big(\dfrac{1}{2}\Big)^2 \\[1em] = \dfrac{1}{2} + \dfrac{3}{4} + \dfrac{1}{4} \\[1em] = \dfrac{2 + 3 + 1}{4} \\[1em] = \dfrac{6}{4} \\[1em] = \dfrac{3}{2}.$

Hence, cos² 45° + sin² 60° + sin² 30° = $\dfrac{3}{2}$.

(iv) cos 90° + cos² 45° sin 30° tan 45°

$= 0 + \Big(\dfrac{1}{\sqrt{2}}\Big)^2 \times \dfrac{1}{2} \times 1 \\[1em] = 0 + \dfrac{1}{2} \times \dfrac{1}{2} \\[1em] = \dfrac{1}{4}.$

Hence, cos 90° + cos² 45° sin 30° tan 45° = $\dfrac{1}{4}$.