Evaluate

Question.

Evaluate

(i) $\frac{\sin ^{2} 63^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\cos ^{2} 73^{\circ}}$

(ii) $\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$

Solution:

(i) $\frac{\sin ^{2} 63^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\cos ^{2} 73^{\circ}}$

$=\frac{\left\{\sin \left(90^{\circ}-27^{\circ}\right)\right\}^{2}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\left\{\cos \left(90^{\circ}-17^{\circ}\right)\right\}^{2}}$

$=\frac{\left\{\cos 27^{\circ}\right\}^{2}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\left\{\sin 17^{\circ}\right\}^{2}}$

$=\frac{\cos ^{2} 27^{\circ}+\sin ^{2} 27^{\circ}}{\cos ^{2} 17^{\circ}+\sin ^{2} 17^{\circ}}=\frac{1}{1}=1$

(ii) $\sin 25^{\circ} \cos 65^{\circ}+\cos 25^{\circ} \sin 65^{\circ}$

$=\sin \left(90^{\circ}-65^{\circ}\right) \cos 65^{\circ}+\cos \left(90^{\circ}-65^{\circ}\right) \sin 65^{\circ}$

$=\cos 65^{\circ} \cos 65^{\circ}+\sin 65^{\circ} \sin 65^{\circ}$

$=\cos ^{2} 65^{\circ}+\sin ^{2} 65^{\circ}=1$