use a half - angle for\n$\tan 157.5^{circ}$\n$\tan 157.5^{circ}=square$\n(simplify your answer, including…

Explanation:

Step1: Determine the half - angle formula

The half - angle formula for tangent is (\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}) or (\tan\frac{\alpha}{2}=\frac{\sin\alpha}{1+\cos\alpha}). Let (\alpha = 315^{\circ}), then (\frac{\alpha}{2}=157.5^{\circ}).

Step2: Find the values of (\sin\alpha) and (\cos\alpha)

We know that (\sin315^{\circ}=-\frac{\sqrt{2}}{2}) and (\cos315^{\circ}=\frac{\sqrt{2}}{2}).

Step3: Substitute into the half - angle formula

Using (\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}), substitute (\alpha = 315^{\circ}): [ \begin{align*} \tan157.5^{\circ}&=\frac{1-\cos315^{\circ}}{\sin315^{\circ}}\ &=\frac{1-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\ &=\frac{\frac{2 - \sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}\ &=-( \sqrt{2}-1)\ &=1-\sqrt{2} \end{align*} ]

Answer:

(1-\sqrt{2})