Question

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°

Answer 1

(i) sin 60^o cos 30^o − cos 60^o sin 30^o
$$\begin{gathered} =\left(\frac{\sqrt{3}}{2}\right)\times \left(\frac{\sqrt{3}}{2}\right)-\left(\frac{1}{2}\right)\times \left(\frac{1}{2}\right)=\frac{3}{4}-\frac{1}{4}=\frac{2}{4}=\frac{1}{2} \\ \mathrm{Also},\sin {30}^{\mathrm{o}}=\frac{1}{2} \end{gathered}$$
∴ sin 60^o cos 30^o − cos 60^o sin 30^o = sin 30^o

(ii) cos 60^o cos 30^o + sin 60^o sin 30^o

​$$\begin{gathered} =\left(\frac{1}{2}\right)\times \left(\frac{\sqrt{3}}{2}\right)+\left(\frac{\sqrt{3}}{2}\right)\times \left(\frac{1}{2}\right)=\frac{\sqrt{3}}{4}+\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{2} \\ \mathrm{Also},\cos {30}^{\mathrm{o}}=\frac{\sqrt{3}}{2} \end{gathered}$$
∴ cos 60^o cos 30^o + sin 60^o sin 30^o = cos 30^o

(iii) 2 sin 30^o cos 30^o
$$\begin{gathered} =2\times \frac{1}{2}\times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2} \\ \mathrm{Also},\sin {60}^{\mathrm{o}}=\frac{\sqrt{3}}{2} \end{gathered}$$
∴ 2 sin 30^o cos 30^o = sin 60^o

(iv) 2 sin 45^o cos 45^o
$=2\times \frac{1}{\sqrt{2}}\times \frac{1}{\sqrt{2}}=1$
Also, sin 90^o = 1
∴ 2 sin 45^o cos 45^o = sin 90^o