simplify the expression by using a double - angle formula or a half - angle formula.\n(a) \\( \\frac { \\sin…

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)