If $\sin (\mathrm{A}+\mathrm{B})=\sin \mathrm{A} \cos \mathrm{B}+\cos \mathrm{A} \sin \mathrm{B}$ and $\cos (\mathrm{A}-\mathrm{B})=\cos \mathrm{A} \cos \mathrm{B}+\sin \mathrm{A} \sin \mathrm{B}$, find the values of (i) $\sin 75^{\circ}$ and (ii) $\cos 15^{\circ}$.
Let $\mathrm{A}=45^{\circ}$ and $\mathrm{B}=30^{\circ}$
(i)
As, $\sin (A+B)=\sin A \cos B+\cos A \sin B$
$\Rightarrow \sin \left(45^{\circ}+30^{\circ}\right)=\sin 45^{\circ} \cos 30^{\circ}+\cos 45^{\circ} \sin 30^{\circ}$
$\Rightarrow \sin \left(75^{\circ}\right)=\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \times \frac{1}{2}$
$\Rightarrow \sin 75^{\circ}=\frac{\sqrt{3}}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}$
$\therefore \sin 75^{\circ}=\frac{\sqrt{3}+1}{2 \sqrt{2}}$
(ii)
As, $\cos (A-B)=\cos A \cos B+\sin A \sin B$
$\Rightarrow \cos \left(45^{\circ}-30^{\circ}\right)=\cos 45^{\circ} \cos 30^{\circ}+\sin 45^{\circ} \sin 30^{\circ}$
$\Rightarrow \cos \left(15^{\circ}\right)=\frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}} \times \frac{1}{2}$
$\Rightarrow \cos 15^{\circ}=\frac{\sqrt{3}}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}$
$\therefore \cos 15^{\circ}=\frac{\sqrt{3}+1}{2 \sqrt{2}}$
Disclaimer: $\cos 15^{\circ}$ can also be calculated by taking $\mathrm{A}=60^{\circ}$ and $\mathrm{B}=45^{\circ}$.
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