84
If $\tan A = 1$ and $\tan B = \sqrt{3}$, then evaluate ; $\cos A \cos B + \sin A \sin B$.
Show SolutionHide Solution↓
$A = 45^\circ, B = 60^\circ$
$\cos A \cos B + \sin A \sin B$
$= \cos 45^\circ \cos 60^\circ + \sin 45^\circ \sin 60^\circ$
$= \frac{1}{\sqrt{2}} \times \frac{1}{2} + \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}$
$= \frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}} = \frac{1+\sqrt{3}}{2\sqrt{2}}$
$\cos A \cos B + \sin A \sin B$
$= \cos 45^\circ \cos 60^\circ + \sin 45^\circ \sin 60^\circ$
$= \frac{1}{\sqrt{2}} \times \frac{1}{2} + \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2}$
$= \frac{1}{2\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}} = \frac{1+\sqrt{3}}{2\sqrt{2}}$