Question

simplify the expression by using a double - angle formula or a half - angle formula.
(a) \\( \frac { \sin ( 10 ^ { \circ } ) } { 1 + \cos ( 10 ^ { \circ } ) } \\)
(b) \\( \frac { 1 - \cos ( 8 \theta ) } { \sin ( 8 \theta ) } \\)

Explanation:

Step1: Recall the half - angle formula

The half - angle formula for \(\tan\frac{\alpha}{2}=\frac{\sin\alpha}{1 + \cos\alpha}\). For part (a), let \(\alpha = 10^{\circ}\).

Step2: Apply the formula

Substituting \(\alpha = 10^{\circ}\) into \(\frac{\sin\alpha}{1+\cos\alpha}\), we get \(\frac{\sin(10^{\circ})}{1+\cos(10^{\circ})}=\tan\frac{10^{\circ}}{2}=\tan5^{\circ}\).

Step3: Recall another half - angle formula

The half - angle formula for \(\tan\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}\). For part (b), let \(\alpha = 8\theta\).

Step4: Apply the formula

Substituting \(\alpha = 8\theta\) into \(\frac{1 - \cos\alpha}{\sin\alpha}\), we get \(\frac{1-\cos(8\theta)}{\sin(8\theta)}=\tan\frac{8\theta}{2}=\tan4\theta\).

Answer:

(a) \(\tan5^{\circ}\)
(b) \(\tan4\theta\)