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Evaluate the following:

Question:

Evaluate the following:

(i) sin 78° cos 18° − cos 78° sin 18°

(ii) cos 47° cos 13° − sin 47° sin 13°

(iii) sin 36° cos 9° + cos 36° sin 9°

(iv) cos 80° cos 20° + sin 80° sin 20°

Solution:

(i) $\sin 78^{\circ} \cos 18^{\circ}-\cos 78^{\circ} \sin 18^{\circ}$

$=\sin \left(78^{\circ}-18^{\circ}\right)$         $[\mathrm{U} \operatorname{sing} \sin A \cos B-\cos A \sin B=\sin (A-B)]$

$=\sin 60^{\circ}=\frac{\sqrt{3}}{2}$

(ii) $\cos 47^{\circ} \cos 13^{\circ}-\sin 47^{\circ} \sin 13^{\circ}$

$=\cos \left(47^{\circ}+13^{\circ}\right)$       $[\mathrm{U} \operatorname{sing} \cos A \cos B-\sin A \sin B=\cos (A+B)]$

$=\cos 60^{\circ}=\frac{1}{2}$

(iii) $\sin 36^{\circ} \cos 9^{\circ}+\cos 36^{\circ} \sin 9^{\circ}$

$=\sin \left(36^{\circ}+9^{\circ}\right)$           $[\mathrm{U} \operatorname{sing} \sin A \cos B+\cos A \sin B=\sin (A+B)]$

$=\sin 45^{\circ}=\frac{1}{\sqrt{2}}$

(iv) $\cos 80^{\circ} \cos 20^{\circ}+\sin 80^{\circ} \sin 20^{\circ}$

$=\cos \left(80^{\circ}-20^{\circ}\right)$            [Using $\cos A \cos B+\sin A \sin B=\cos (A-B)$ ]

$=\cos 60^{\circ}=\frac{1}{2}$

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