use a half - angle formula to find the exact value of the following expression.\n\ntan 157.5°\n\n\ntan…

Explanation:

Step1: Determine the formula

The half - angle formula for tangent is (\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}) when (\frac{\alpha}{2}) is in the second quadrant (since (157.5^{\circ}) is in the second quadrant, (\tan157.5^{\circ}<0)). Here (\frac{\alpha}{2}=157.5^{\circ}), so (\alpha = 315^{\circ})

Step2: Substitute (\alpha = 315^{\circ}) into the formula

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

[ \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})